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A COURSE OF
DIFFERENTIAL
GEOMETEY
Oxford University Press
London Edinburgh Glasgow Copenhagen
New York Toronto Melbourne Cape Town
Bombay Calcutta Madras Shanghai
Humphrey Milford Publisher to the UNIVERSITY
A COURSE OF
DIFFERENTIAL
GEOMETRY^
BY THE LATE
JOHN EDWARD CAJMPJBELL
M.A. (OXON.), HON.D.SC, (BELFAST), F.R.S.
FELLOW OF HERTFORD COLLEGE, OXFORD
PREPARED FOR THE PRESS WITH
THE ASSISTANCE OF
E, B. ELLIOTT
M.A., F.R.S., EMERITUS PROFESSOR
OXFORD
AT THE CLARENDON PRESS
1926
Printed in England
At the OXFORD UNIVERSITY PRESS
By John Johnson
Printer to the University
PREFACE
MY father .had spent most of his spare time since
the War in writing this book. Only two .months
before his death, while on our summer holiday in 1924,
he had brought some of the chapters with him, and
sent off the final draft of them to the Clarendon Press.
Even on these holidays, which he greatly enjoyed, we
were all accustomed to a good deal of work, and it
was an unexpected pleasure to find that with these
once dispatched to the press he took an unusually
complete holiday.
While rejoicing that he was so far able to com-
plete the book, we are sorry that a last chapter or
appendix in which he was greatly interested was
hardly begun. Apparently this was to deal with the
connexion between the rest of the book and Einstein's
theory. To the mathematical world his interest in
this was shown by his Presidential address to the
London Mathematical Society in 1920 to his friends
by the delight he took on his frequent walks in trying
to explain in lucid language something of what
Einstein's theory meant.
vi PREFACE
We cannot be too grateful to Professor Elliott,
F.R.S., s,n old friend of many years standing, for
preparing the book for the press and reading and
correcting the proofs. No labour has been too great
for him to make the book as nearly as possible what it
would have been. And the task has been no light one.
We should like to thank the Clarendon Press for
their unfailing courtesy and for the manner in which
the book has been produced.
J. M. H. C.
Christmas 1925.
EDITOR'S NOTE
MY dear friend the author of this book has devoted
to preparation for it years of patient study and inde-
pendent thought. Now that he has passed away, it
has been a labour of love to me to do my best for him
in seeing it through the press. As I had made no
special study of Differential Geometry beforehand, and
was entirely without expertness in the methods of
which Mr. Campbell had been leading us to realize
the importance, there was no danger of my converting
the treatise into one partly my own. It stands the
work of a writer of marked individuality, with rather
unusual instincts as to naturalness in presentation.
A master's hand is shown in the analysis.
Before his death he had written out, and submitted
to the Delegates of the University Press, nearly all
that he meant to say. An appendix, bearing on the
Physics of Einstein, was to have been added ; but
only introductory statements on the subject have been
found among his papers. Unfortunately finishing
touches, to put the book itself in readiness for printing,
had still to be given to it. The chapters were numbered
viii EDITOR'S NOTE
in an order which, rightly or wrongly, is in one place
here departed from, but they stood almost as separate
monographs, with only a very few references in general
terms from one to another. To connect them as the
author would have done in due course is beyond
the power of another. The articles, however, have
now been numbered, and headings have been given to
them. Also some references have been introduced.
The text has not been tampered with, except in details
of expression ; but a few foot-notes in square brackets
have been appended.
E. B. E.
TABLE OF CONTENTS
CHAPTER I
TENSOR THEORY
PAGE
The n-way differential quadratic form (1) 1
The distance element. Euclidean and curved spaces (2) 2
Vectors in a Euclidean space which trace out the space of a
form (3) 4
Christoffel's two symbols of three indices (4) . v ~r*~~^ . . 6
Some important operators (5) \^^^ ^^"'^'
Conclusions as to derivatives of ajj., a ll> , and i lg#~rt'(6) . ****T . 8
Tensors and tensor components defined (7j^X*C .... 9
The functions a^ and a lk aie tensor components (8) ,, . . . 10
Expressions for second derivatives when a^dx^dx^ = o! ^dx {dx ^ (9) 11
Tensor derivatives of tensor components are tensor components (10) 12
Rules and definitions of the tensor calculus (11) .... 14
Beltrami's three differential parameters (12) 16
Two associated vector spaces. Normals to surfaces (13) . . . 17
Euclidean coordinates at a point (14) 18
Two symbols of four indices which are tensor components (15) . 19
A four-index generator of tensor components from tensor com-
ponents (16) 21
Systems of invariants (17) .22
An Einstein space, and its vanishing invariants (18) ... 23
CHAPTER II
THE GROUND FORM WHEN n = 2 ''
Alternative notations (19) 25
An example of applicable surfaces (20) 26
Spherical and pseudo-spherical surfaces. The tractrix revolution
surface (21) 27
X TABLE OF CONTENTS
PAGE
Ruled and developable surfaces. The latter applicable on a
plane (22) 28
Elliptic coordinates (23) 31
The invariant K (24) 32
Determination of a ^ such that A (<, t//>) = (25) .... 34
Reduction of a ground form when K is constant (26) . . . o 34
The case of A (JST) (27) 36
The case when A 2 K and A K are functions of K (28) ... 37
Conditions for equivalence in the general case (29) .... 38
The functions called rotation functions (30) 39
Integration of the complete system of 24 (31) . . . . 40
CHAPTER III
GEODESICS IN TWO-WAY SPACE
Differential equation of a geodesic ( 32) . ^S .... 42
Another form of the equation (33) . s . . , " .43
Condition that orthogonal trajectories be geodesic^ {34) . . 44
Geodesic curvature (35, 36) ". 45
Polar geodesic coordinates (37) . - 47
Recapitulation. Parallel curves (38) 49
Notes regarding geodesic curvature (39, 40) 51
Integration of geodesic equations when K\* constant (41) . . 54
Focal coordinates (42) 55
Explicit expressions for symbols [ikj] and for K (43) ... 55
Liouville's special form (44) 58
Null lines. Complex functions of position (45) . . . .58
Conjugate Harmonic Functions. Mapping on a plane (46) . . 60
CHAPTER IV
TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
A quaternion notation (47) 62
Introduction of new fundamental magnitudes and equations (48) . 63
Connexion of the magnitudes with curvature (49) .... 64
The normal vector determinate when the functions Qy are
known (50) 65
Reference to lines of curvature. The measure of curvature (51) . 68
Tangential equations. Minimal surfaces (52) 69
Weingarten or W surfaces (53) 71
TABLE OP CONTENTS XI
PAGE
An example of W surfaces (54) 72
The spherical and pseudo-spherical examples (55) . ..... 7S
Reference to asymptotic lines (56) 75
Equations determining a surface (57) 78
The equation for the normal vector in tensor form (58) ... 79
Introduction of a new vector (59) 80
Orthogonally corresponding surfaces (60) 81
Recapitulation (61) 82
Relationship of surfaces z and f (62) 83
Association of two other surfaces with a ^-surface (63, 64) . . 84
CHAPTER V
DEFORMATION OF A SURFACE, AND CONGRUENCES
Continuous deformation of a surface (65) 86
A vector of rotation (66) 87
Geometrical relationship of surfaces traced out by certain vectors
(67,68) 88
A group of operators, and a system of twelve associated surfaces
traced out by vectors (69, 70) 89
The twelve surfaces form three classes of four (71) .... 91
A case in which one surface is minimal (72) 92
Congruences of straight lines (73) 93
Focal planes and focal points of a ray (74) 94
Limiting points. The Hamiltonian equation. Principal planes (75) 96
Principal surfaces, and the central surface (76) . . .97
The focal surface (77) ... 98
Rays touch both sheets of the focal surface. The congruence of
rays of light (78) 98
Refraction of a congruence. Malus's theorem (79) .... 100
The Ribaucourian congruence (80) 101
The Isotropic congruence. Ribaucour's theorem (81) . . . 102
W congruences (82) 103
Congruence of normals to a surface (83) 104
Reference to lines of curvature (84) 105
Tangents to a system of geodesies (85) 105
Connexion of W congruences which are normal with If surfaces (86) 106
Surfaces applicable to surfaces of revolution, and W normal
congruences (87) 107
.Surfaces of constant negative curvature (88) 108
Xll TABLE OF CONTENTS
CHAPTER VI
CURVES IN EUCLIDEAN SPACE AND ON A SURFACE.
MOVING AXES
PAGE
ferret's formulae. Rotation functions ( 89) r 110
Codazzi's equations (90) Ill
Expressions for curvature and torsion (91) 113
Determination of a curve from Serret's equations (92) . . .114
Associated Bertrand curves (93) 116
A curve on a surface in relation to that suiface (94) . . .117
Formulae for geodesic torsion and curvature (95) .... 120
Surfaces whose lines of curvature are plane curves (96j . . . 121
Enneper's theorem (97) 125
The method of moving axes (98) 125
Orthogonal surfaces (99) 127
CHAPTER VII
THE RULED SURFACE
Unit orthogonal vectors ( 100) 129
The ground form and fundamental magnitudes (101) . . . 130
Bonnet's theorem on applicable ruled surfaces (102) . . . 131
Ground forms applicable on a ruled surface (103) . . . .133
Case of applicability to a quadric (104) 135
Special ground forms. Binomials to a curve. Line of stric-
tion(105) 136
Constancy of anharinonic latios. Applicable ruled surfaces and
surfaces of Revolution (106) 138
Surfaces cutting at one angle all along a generator (107) . . 139
The ruled surfaces of an isotropic congruence (108) . . .141
CHAPTER VIII
THE MINIMAL SURFACE
Formulae and a characteristic propeity ( 109) .... 143
Reference to null lines. Stereographic projection (110) . . . 145
The vector of a null curve (111) 147
Self-conjugate null curves. They may be (1) unicursal, (2) al-
gebraic (112) 147
TABLE OF CONTENTS Xlll
FAGE
Generation of minimal surfaces from null curves. Double minimal
surfaces (113) t . 149
Henneberg's surface (114) 151
Lines of curvature and asymptotic lines on minimal surfaces (115) . 152
Associate and adjoint minimal surfaces (116) 153
CHAPTER IX
THE PROBLEM OF PLATEAU AND CONFORMAL
REPRESENTATION
The minimal surface with a given closed boundary ( 117) . . 155
The notation of a linear differential equation of the second order
with three singularities (118) 157
Conforinal representation on a triangular area (119) . . . 158
The w-plane or part of it covered with curvilinear triangles (120) . 161
Consideration of the case when triangles do not overlap (121) . 163
Case of a real orthogonal circle as natural boundary (122) . . 165
Fundamental spherical triangles when there is no natural boun-
dary (123) 166
Summary of conclusions (124) 168
Representation of the o>plane on a given polygon (125, 126) . . 168
CHAPTER X
ORTHOGONAL SURFACES
A certain partial differential equation of the third order ( 127) . 172
A solution led to when functions satisfying a set of three equations
are known (128) 174
The vector q(Xq~ l , where OK is a vector and q a quaternion (129) . 174
Passage from set to set of three orthogonal vectors (130) . . 175
Rotation functions (131) 177
A vector which traces out a triply orthogonal system (132) . . 178
Lines and measures of curvature (133) 179
Linear equations on whose solution depends that of the equation
of the third order (134) 181
Synopsis of the general argument (135) 182
An alternative method indicated (136) 184
Three additional conditions which may be satisfied (137) . . 185
^Orthogonal systems from which others follow by direct operations
(138) 186
XIV TABLE OF CONTENTS
CHAPTER XI
INFERENTIAL GEOMETRY IN n-WAY SPACE
PAGE
Geodesies in n-way space ( 139) 188
Geodesic polar coordinates and Euclidean coordinates at a point
(140) 190
Riemann's measure of curvature of n-way space (141) . . . 193
Further study of curvature. The Gaussian measures for geodesic
surfaces. Orientation (142) - 194
A notation for oriented area (143) 198
A system of geodesies normal to one surface are normal to a system
of surfaces (144) 199
The determination ot surfaces orthogonal to geodesies and of
geodesies orthogonal to surfaces (145) 203
A useful reference in (n + 1)- way space (146) 205
Geometry of the functions fi t -fc (147) 207
The sum of the products of two principal curvatures at a point
(148) 209
Einstein space (149) 211
An (n+ l)-way Einstein space surrounds any given n-way space
(150-4) 212
CHAPTER XII
THE GENERATION OF AN (w + l)-WAY STATIONARY EINSTEIN
SPACE FROM AN n-WAY SPACE
Conditions that the (n + l)-way Einstein space surrounding a given
n-way space be stationary ( 155) 220
Infinitesimal generation of the (n-f-l)-way from the n-\\ay form
(156) 222
Restatement and interpretation of results (157) .... 225
A particular case examined when n = 2 (158) 226
General procedure in looking for a four- way stationary Einstein
space (159) 228
Conclusions as to curvature (160) ....... 229
CHAPTER XIII
n-WAY SPACE OF CONSTANT CURVATURE
Ground form for a space of zero Riemann curvature ( 161) . . 231
Ground form for a space of constant curvature for all orientations
(162) 232
TABLE OF CONTENTS XV
PAGE
Different forms for these spaces (163) 234
Geodesic geometry for a space of curvature +1 (164) . ., . 236
Geodesies as circles (165) . .' 237
Geodesic distance between two points (166) 238
Coordinates analogous to polar coordinates (167) .... 239
The three-way space of curvature 4- 1 (168) 240
The geometry of the space (169) 241
Formulae for lines in the space, and an invariant (170) . . . 243
Volume in the space (171) 246
An n-way space of constant curvature as a section of an extended
Einstein space (172) 247
CHAPTER XIV
n-WAY SPACE AS A LOCUS IN (w + l)-WAY SPACE
A space by which any n-way space may be surrounded ( 173) . 250
Curvature properties of this surrounding space (174, 175) . .251
A condition that the surrounding space may be Euclidean (176) . 253
Procedure for applying the condition when n >2 (177) . . . 255
The n-way space as a surface in the Euclidean space when this
exists (178, 179) 255
INDEX 259
CHAPTER I
TENSOR THEORY
1. The n-wa,y differential quadratic form. Let us
consider the expression
a ik dxidx k , \-= 1 ...n (1.1)
which is briefly written for the sum of r^ such terms, obtained
by giving to i, k independently the values 1, 2, ... n. If, for
instance, n = 2, the expression is a short way of writing
a n dxl + 2ct l2 dj' l dx 2 + a 2 ^dx^ ;
for we are assuming that
tfc=fci- (*- 2 )
Let us also denote by a ih the result of dividing by a itself
(\) i+1 * times the determinant obtained by erasing the row
and the column \vhi cli contain a in the determinant
a =
(1.3)
The coefficients a ik ... are at present arbitrarily assigned
functions of the variables x l ...^' n> limited only by the
condition that a is not zero.
When we are given the coefficients a ik as functions of their
arguments, there must exist r functions
X l ...X r , r = -in(?M-l),
of the variables a^ ... x n , such that
dXl + ...+ dXl = a ik dx 4 dx k . (1.4)
The differential equations which will determine these
functions are %g ^%
= a ik . (1.5)
*i ^ x k
Just as in the expression a ik dx t dx k the law of the notation
is that, whenever a suffix, which occurs in one factor of
TENSOR THEORY
a product, is repeated in another factor^ the sum of all such
products is to be taken, so here the above differential equation
is the short way of writing
As there are just as many unknown functions as there
are differential equations to be satisfied, we know that the
functions X l ... X r must exist. The actual solution of this
system of differential equations is, however, quite another
matter, and questions connected with the solution form a chief
part in the study of Differential Geometry.
2. The distance element. Euclidean and curved spaces.
If we regard x lt ..x n as the coordinates of a point in an
7i-way space, then, X 1 ... X r being functions of y\ ...x n . we
may regard this space as a locus in r-way Euclidean space ;
and we may regard dk as the distance between two neighbour-
ing points x l . . . x n an d x 1 + dx l ...x n + dx n , where ds is
defined by ds z = (f{k d Xi dx k . (2.1)
Thus, if n = 2, the two-way space given by
ds 2 = atffdxidxfr
lies within our ordinary Euclidean space, and it is with this
space that Differential Geometry has hitherto been chiefly
concerned.
If 7i = 3, the * curved * three-way space lies, in general,
within a Euclidean six-way space. If, however, the coefficients
a^, instead of being arbitrarily assigned functions of their
arguments x lt o/ 2 , # 3 , satisfy certain conditions, the Euclidean
space may be only a five- way space, or even only a four- way
space. In yet more special cases the three-way space may
not be * curved 1 at all, but only ordinary Euclidean space
with a different coordinate system of reference.
If n = 4, the curved four-way space lies, in general, within
a Euclidean ten- way space, and so on.
We know what a curved two-way space within a Euclidean
three-way space means, being a surface : but what does
a ' curved ' three-way space mean ? We have not, and we
DISTANCE ELEMENT. EUCLIDEAN AND CURVED SPACES 3
cannot have, a conception of a four- way space, Euclidean or
otherwise, within which the three-way space is to be curved.
But by thinking of the geometry associated with the form
cfe 2 = a u dxl + 2<f 12 davir 2 + ff 22 <fo5 ( 2 2 )
we say that it is that of a curved two-way space ; and we
know that it is, in general, different from the flat Euclidean
plane geometry associated with the form
d*> = dxl+dxl. (2.3)
We can distinguish these two geometries without any
leference to the Euclidean three-way space, or any other
three-way space. This distinction we, with our knowledge
of a three-way Euclidean space, characterize by saying that
the first space is curved and the second flat, or Euclidean.
This is what we mean when we say that the space given by
ds* = (tfadxidxje (2 4)
is, in genera], a curved space, whilst that given by
d# = <Y.Y? + ...+<W:* (2.5)
is a flat space. We shall find that a geometrical property
will be associated with a curved space, which will distinguish
it from a flat space.
If we have no real knowledge of a space of more than
three dimensions, we have at least no knowledge that it does
not exist: and, by analogy from our knowledge both of
a two-way space and a three-way space, we are able to make
use of the ideas of higher space to express analytical results
in an interesting form.
The space in which wo live may, or may not, be flat or
Euclidean. Up till quite recently it has been assumed to be
flat, and the geometry which has been built up has been
that associated with the form
The geometry which we wish to know about to-day would
be that associated with the form
ds 2 = afodxtdxfa \ = 1 ... 4,
where x...x are functions of the three variables which
4 TENSOR THEORY
locate an event in space, and a fourth variable which locates
it in time.
The geometry of Euclidean space is much simpler than the
geometry associated with the more general form, and its
properties have been more studied. It may therefore be of
advantage, at least in some ways, to regard the form
ds* = aftdxidxje (2.1)
as that of an ft- way locus in a tiat r-way space, although r is
generally a much larger number than n.
3. Vectors in a Euclidean space which trace out the
space of a form. Let i' \ i" , i'" ... be r unit vectors in the
Euclidean space and let y and z be vectors given by
What we call the scalar product of the two ^vectors y and
z is denoted by yz and defined by
y^ + y' Z ' + y"z"+... =0. (3.2)
The cosine of the angle between the vectors is defined as
-yz
and may be written _^_ . (3 . 3)
\yy zz
We shall generally write yy as 2/ 2 , but we must remember
then that the root of y 1 is not y.
The numbers y', y" ... are called the components of the
vector y : they are ordinary scalar numbers.
Now let z be a vector whose components are functions of
the n parameters x l ... x n . Denoting the derivative of z with
respect to x r by z r , we have dz = z p dx p in the notation we
have explained, which is the foundation of the Tensor
Calculus. We therefore have
dzdz = ZiZkdXidXfr. (3.4)
VECTORS IN A EUCLIDEAN SPACE 5
The vector z traces out an n-w&y space within the Euclidean
r-way space, and in this ?i-way space the element of ' length '
is given by efc = -cfe<fe; (3.5)
and therefore, if we take
a ik = ~Wk> (3.6)
we have cfes
We say that cite is an element in this space, and we notice
that an element has direction as well as length. The element
is localized at the extremity of the vector z ; the element lies
in the w-way space, but the vector lies in the r-way Euclidean
space.
The direction cosines of the element in the r-way space are
, dx n dx p
**-*' **-*"- (3 * 7)
We write fp = />
s 7 >
els
and we speak of J , 2 , ... g as the direction cosines of the
element in the ?i-way space given by, or associated with,
ds 2 = a
The upper affixes in g l ... * have, of course, no implication
of powers as in ordinary algebra. The notation introduced is
in accordance with that of the tensor calculus which we are
leading up to. In accordance with that calculus we ought
to write the variables x l . . . x n as a; 1 ... x n , but we do not do
so, as the notation x l .. f x )l is at present too firmly fixed
perhaps.
If CD is the angle between two elements, drawn through
the extremity of 0, whose direction cosines with respect to
the n- way space are i 2 tn
respectively,
q = u pq l '*] (1 - (3.8)
6 TENSOR THEORY
It should be noticed that ^pq^rj^ means precisely the same
thing as fl^^V' Repeated suffixes are called dummy suffixes
and'can be replaced by any other dummy suffixes. The chief
rule that we need to follow is not to use the same dummy
more than twice in an expression containing a number of
factors. *
It should be noticed that the angle, for which we have
found an expression, is that between two elements drawn
through the same point, viz. the same extremity of the
vector z. We have no expression for the angle between two
elements at different points in our 7i-way space. This is
something that distinguishes the geometry connected with
the form ds 2 a^dx^lx^ from the geometry of Euclidean
space.
4. ChristofTel's two symbols of three indices. Let
. (4.1)
This is the definition of Chris toffel's three-index symbol of
the first kind. It is exceedingly important in the theory
of differential geometry. The first two suffixes are inter-
changeable. We may write it sometimes in the form T t
when we regard i and k as fixed suffixes.
Since fl<fe = -^i
we see that (ikl) = ~^^, (4 2 )
, Wz
where z ijf = .- v
ll{ dx.^rfr
We introduce the symbol e\ to denote zero if i and k are
unequal and unity if i and k are equal. We do not write
J as equal to unity, for by our convention
e{=j+e!| + ... + = n.
In employing dummy suffixes it is best to employ a letter
to which we have not attached a definite connotation.
From the property of determinants and their first minors
we see that a it 0kf = ^ m (i . 3)
CHRISTOFFEL S TWO SYMBOLS OF THREE INDICES
Let {ikj} =o#(arf); (4.4)
then {ikj} is Christoffel's three-index symbol of the second
kind. The first two suffixes are interchangeable and it may
be written TJ when we regard i and k as fixed suffixes.
We have at once (/&j) ^ ajt {ikt}. (4 . 5)
5. Some important operators. Even already we have
come across a number of functions of the vaiiables which we
denote by integers attached to a certain letter. Thus we
have the fundamental functions denoted by u^...; we have
the direction cosines denoted by 1 ... n , and the functions
a ik ....
More generally we may have a number of functions of the
variables, say 6, <, >//, ... and we may form a function of
6, 0, ^Jr, ... and their derivatives with respect to the variables.
It may be that the function thus arrived at may bs denoted by
where a, j8, ... are integers of the upper row, upper integers
we call them, and a, b y ... are lower integers. These integers
may take independently any of the values 1, 2, ... u and thus
indicate how the function 7', "" is formed. The number
-/ a, 5, ...
of the upper integers is not necessarily equal to the number
of the lower integers. It may be that there are no integers
in the upper row, or none in the lower, or even none in either.
We shall come across many functions which may be ex-
pressed in this manner, and we have come across some.
In connexion with functions which are expressed in the
above form there are n operators which are of fundamental
importance in tensor theory. These operators may be written
1,2, 3,...rc,
where p denotes the operator
and where (') denotes the operation of substituting t for A,
A.
TENSOR THEORY
A being any upper integer, and where (*) has a similar
meaning with respect to a lower integer.
(5.3)
the which occurs on the right is a dummy suffix, and thus,
for instance,
We notice that the definite integers 1, 2, ... ?i are not
dummies, and we should avoid the use of 71 as a dummy.
-x - - /e e ^
We write 1 a- P =l> 1 a ( 5 5 )
By aid of the symbolism thus introduced we can avoid
a prolixity which would otherwise almost bar progress. A
very little practice will enable one to use this symbolism
freely, and when necessary to express the results explicitly.
6. Conclusions as to derivatives of a ijc) a* 1 *, and
We see from the definition that
; (6.1)
and therefore the operator p annihilates each of the functions
a ik , which, of course, could have been written T ik .
We have a a ^ = ;
and therefore
/fc r a* 7 + a u (tpk) + a u (kpt) = 0.
dx p
It follows that
a**a tk <- (fi + a** {tpq}+aH {kpi} = ;
that is, a ty +a < {pg} +a {ipi} =0. (6 .
CONCLUSIONS AS TO DERIVATIVES 9
It follows that the operator p also annihilates each of the
functions a ik .
By the rule for the differentiation of a determinant
= a (tl )( l (ptq) + a a!"l (qtp)
- 2a{i>tp],
<* i i
or - </2 a*{ptp}. (6.3)
This formula will be required later. [It should be re-
membered that the symbol on the right stands for the sum of
n symbols, with p 1, 2, ... a. ]
7. Tensors and tensor components defined. We must
now explain what is meant by a tensor. We have seen how
functions denoted by
r a > Py
(iy 6. * * *
may be derived from functions $,</>, ty ... and their derivatives
with respect to X 1 ... x n . The different functions obtained by
allowing the integers to take all values from 1 up to n are
called components of the set.
Suppose that we transform to ne\v variables a;'. ...x' n , and
that & denotes the expression of 6 in terms of the new
variables, and that (/>', ty' ... have similar meanings. Suppose
further that r/ya',0',
/ / i./
M (i f) ... * *
are functions formed from 0', X , ^', ... and their derivatives
with respect to the new variables x\ ...x }l by exactly the
same rules as the functions
rot, A...
a, b, ...''
were formed from 0, 0, \/r, and their derivatives with respect
10 TENSOR THEORY
We say that 7^6,' '!.'
are components of a tensor if
TV ',', ... __ jta^ ^ ^V^V nr*,0,...
* ;*,... -5^,a*v" ^ >V" ft ' 6 ""'
Notice that the integers on the left are not dummies but
that the integers a, )8, ... a } 6, ... on the right are, Notice
also that the above equation must hold for all values of the
integers on the left if the expressions
r ,*,...
a, b, ...
are to bo tensor components.
This is the formal definition : we shall immediately come
across examples of ten&ors which will illustrate the definition.
8. The functions a^ and a ijc are tensor components.
If we transform to new variables x\ ... x' n , the expression for
the square of the element of length must remain unaltered in
magnitude though its form may change We therefore have
a pq dx p dx q = "\ndx\dx' p
and so = " <'>
Thus the functions a ik ... satisfy the condition for being
tensor components.
Again from the fundamental equality
,
we have a na ;/ = a
Notice that q and X are no longer dummy suffixes in this
<>x
equality, Multiply across by a' KS ^~ , then we have
8
P t
-r = a >
'
FUNCTIONS a ik AND a lk ARE TENSOR COMPONENTS 11
The'exprossion on the right hand of this equality is
**V ^ _ ^s ^r _ ^V _ r _ -p . .
' ~~ ' ~~ ~ ~ M '
therefore a lia (a r l> - a'** ^f r-^ 2 ) = 0.
jq \ <)xl)x/
and
This equation holds for all values of p, </, and r, and there-
fore, as the determinant a is not zero, wo must have
It follows that the functions a* k ... also satisfy the condition
of being tensor components.
9. Expressions for second derivatives when
We have z' r =
where z f is the expression of z in terms of the new variables,
and %' denotes ^ and z' a denotes ^ , ~ . . It follows,
? Zx' r W * p <>Mq
since by (4 . 2) (pqr) 1 = -^2%, that
Notice, that we see, from this equation, that Chris toffel's
three-index symbols of the first kind do not satisfy the con-
dition of being tensor components.
Multiply across by af r * ^ * , and we have
B
- n
-a
_^- 7 - J> ^ ___ x4 ___
"But, by (8. 2), a' = a'*
12 TENSOR THEORY
and therefore the right-hand member of this equation becomes
We therefore have the fundamental formula in the trans-
formation theory
Similarly we have
,, ^ da' a a; 7 .
J. (9.4,
10. Tensor derivatives of tensor components are tensor
components. We must now show that the operators
1,2, ...n,
when applied to any tensor components, generate other tensor
components.
Let MS L.... X'...
and assume that jT** ^ '" ... are tensor components.
We have >jv >*',.. = T*'^~' MN
JL a', b',... JL a, b, ... " IV
which we briefly write 2 T/ = TJUN.
Expanding -j-M, using the for
dx p'
tion theory which have been obtained,
Expanding -j-M, using the formulae of the transforma-
and therefore
(10.1)
TENSOR DERIVATIVES OF TENSOR COMPONENTS 13
Similarly we have
and therefore
p' N = (^-/^}( l lt }-{^p't}'( l lt ))jf. (10.2)
^X ^\
Now T=--T
i
We have written x , a -rJ- . . . simply as M y but we must
o T* Q 3* i.t
note that Jlf has the upper integers a, 6, ... (as well as the
lower integers a', &', ...) and that the upper integers in M are
the same as the lower integers in T.
Similarly we note that the lower integers in N are the
upper integers in T.
It follows that
if we remember that these lower integers in N and upper
integers in T are just dummies.
We have similarly
M{nql] (') T = T{tq\] (*)Jf, (10.5)
M{w't}'fyT=T{V*}'(D*I> (^.7)
and therefore $>' (TMN) = ^ MNqT.
That is, p'T' = qT^~ ^~ ... r-A^-^ -^-' ..., (10.8)
and therefore '7^ a ^'--- 1 t . are tensor components.
J. Ji Q 0) , , .
This is a very important theorem in the tensor calculus.
It is the rule of taking what we call the tensor derivative
14 TENSOR THEORY
and we see that the tensor derivative of a tensor component
is a tensor component. We denote the p derivative by
T::!;;:. P - 00.9)
11. Bules and definitions of the tensor calculus. We
have now proved the most important theorem in the tensor
calculus: its proof depended on the transformation theorems.
These theorems, having served their purpose, disappear, as it
were, from the calculus.
There are some simple rules of the calculus which we now
consider.
The product of two tensors is a tensor whose components
are the products of each component of the first and each
component of the second tensor. The upper integers of the
product are the upper integers of the two factors, and the
lower integers of the product are the lower integers of
the two factors.
Two tensors of the same character that is, with the same
number of each kind of integers, upper and lower can be
added, if we take together the components which have the
same integers. They can also be combined in other ways, as
we shall see.
We form the tensor derivative of the product of two tensors
by the same rule as in ordinary differentiation.
The tensors a^ and a ift are called fundamental tensors.
We have seen that they have the property of being annihilated
by any operator jp. As regards tensor derivation they there-
lore play the part of constants.
The symbol e^ satisfies the definition of a tensor. It also
is called a fundamental tensor.
Any tensor, formed by taking the product of a tensor and
a fundamental tensor, is said to be an associate tensor of the
tensor from which it is derived.
Suppose that 2 a be '" * s an y tensor. The tensor itself
is the entity made up of all its components, formed by allow-
ing a, /?, y, ..., a, 6, c, ... to take all integral values from
1 up to n. Suppose now, that instead of taking all the
RULES AND DEFINITIONS OF THE TENSOR CALCULUS 15
components, we take those in which one of the upper integers,
say a, is equal to one of the lower integers, say^fc. The
entity we thus arrive at will be a tensor. For
The tensor thus arrived at is denoted by
nrfp, 0,7. ...
* ,P,c, ...
1 'A 4- u 7^,0,7,...
and is said to be y a> &> c>
contracted with respect to a, 6.
We can contract a tensor with respect to any number of
upper integers and an equal number of lower integers.
^l^r(X,
If we take the tensor J[ a b '
an associate tensor would be
a 7^,
e a -/ a, 6
and we might write this y a ' & >
and as it is contracted with respect to two upper integers and
r^
&
^7"fft/3 ^Trt(X/3
So we may write a^tf / pg ns y
We shall often use this contraction when we are consider-
ing associate tensors
The rank of a tensor is the number of integers, upper and
lower, in any component. When the rank is zero the tensor
is an invariant. When the rank is even we can form an
associate tensor which will be an invariant. When the rank
is odd we can form an associate tensor of rank unity. When
the rank is unity the tensor may be said to be a vector in
the 7i- way space : a contravariant vector if the integer is
an upper one, a covariant vector if the integer is a lower
one. But it must be carefully noticed that when we think of
a- vector in the flat r-way space, we are thinking of the word
vector in a different sense, Thua the vector z which traces
16 TENSOR THEORY
out the /i- way space is not an invariant, but rather the entity
of r invariants, and so as regards the derivatives of z. In
the r-way space they are all vectors, hut the coefficients of
the vectors i', i" ... come under the classification of tensors.
If we bear this distinction in mind we shall not be misled,
and we may gain an advantage by combining the two
notions. It is a useless exaggeration of the great advantages
of the tensor calculus to ignore the calculus of Quaternions.
Wo certainly cannot afford to give up the aid of the directed
vector notation in the differential geometry of flat space
within which lies our vi-way curved space.
12. Beltrami's three differential parameters. If we take
any function U of the variables, then
u,,u t , ..u n
will be tensor components. The tensor derivative of an
invariant is just the ordinary derivative; 4ind therefore the
above functions are just the same as
U. l9 U. 99 ...U. n . (12.1)
[For the notation see (5 . 5) and (10 . 9).]
But if we take the second tensor derivatives we come
across different functions from the ordinary second derivatives.
These second tensor derivatives we denote by U . ik ... where
0-. tt =Z7- tt -{ifc}tf,. (12.2)
These we have proved are tensor components ( 10), whereas
the ordinary second derivatives U ik are not. It would be
a useful exercise to prove that the functions U.^... are
tensor components : it might make the general theorem,
whose proof is rather complicated, more easily understood.
The square of the tensor whose components are U l ... U n
is a tensor whose components are U^U^ C . If we form the
associate tensor a^U^U^ we have an invariant which is
denoted by A ([/), so that
&(U) =a ik U { U k . (12.3)
This is Beltrami's first, differential parameter.
Similarly by forming the tensor which is the product of
BELTRAMIS THREE DIFFERENTIAL PARAMETERS 17
the two tensors whose components are t^ ... U n and V l ... V n ,
and taking the associate tensor a ik U i V k , we have B.eltrarni's
* mixed* differential parameter
A (17, V) = a ik U 4 V k . (12.4)
We. also have Beltrami's second differential parameter
& 2 (U) = a? k U. ik . (12.5)
Clearly all these ' difFerential parameters ' as they are called
are invariants. They are of great utility, as we shall find, in
differential geometry.
13. Two associated vector spaces. Normals to surfaces.
Returning now to the vector z, whose extremity traces out
the n-way space within the flat r-way space, we have,
see (5.2) and (12.2),
*-M = *<ft-{'^}**- t 13 - 1 )
Clearly the components of this vector z . ^ are tensor com-
ponents.
We have ^/*,(>i+ 1) vectors z. ik and we have n vectors z i ;
as these (%n(n+ 1) +n) vectors all lie in a %n(n+l) flat
space there must be n linear equations connecting them.
These vectors all depend on the parameters x l9 ..x nt and we
may regard them as all localized at the extremity of the
vector 0.
Now, see 4,
Z ' ih z p = z iJe Z p ~
tp { ikt } = 0. (13.2)
We thus see that the vector z . i1c is perpendicular to every
element in the ?i-way space drawn through the extremity of z.
Let one of the n equations which connect the vectors
*-ifc' *; bo b ik z -ik + b t-t= >
where b ik ...b t ... are scalars. Multiply the equation by
and take the scalar product : then, since
we have
that is, t } t a tp ~
18 TENSOR THEORY
and therefore, since the determinant a cannot be zero, we have
6j = 6 a = .. = 0. (13.3)
It follows that the n linear equations connect the vectors
z> ih ... only.
At any point of the ?i-way space, therefore, there are
n vectors z l ...z n generating a flat ii-way space; and there
are -|/i(/i-f-l) vectors .^., only %n(nl) of which are
linearly independent, and these generate a %n(nl) Hat
space. These two flat spaces, associated with the point
x l ... x n , are such that every element in the one space, drawn
through the extremity of 0, is perpendicular to every element
in the other space, drawn through the extremity of z.
Thus when n is equal to ?, as it is in ordinary differential
geometry, the vectors 3 . u , z . 12 , z . 22 (13.3)
are parallel to the normal at the extremity of z which traces
out the surface we are concerned with.
14. Euclidean coordinates at a point. Associated with
every point x l ...x n we have a special sjstem of coordinates
which we call the Euclidean coordinates of the point. They
are very helpful in proving tensor identities, which without
their aid would prove very laborious.
At the point under consideration a ik ... (ikj) ... are constants.
Let another set of constants be defined by
a ik = b it b ki> b ik = 6 M' i 14 - 1 )
and then another set by
(ikj) = b jt c tik , c m = c (M , (14.2)
and consider the transformation scheme
We have
z J = z 'p( h PJ + c riq
*ik = z \ ( b \i + c \
and therefore at the point
a tt
(ikj) =
EUCLIDEAN COORDINATES AT A POINT 19
Now t/ tt = I'^k = eJA;^/,,
and the determinant a is equal to the square of thje deter-
minant 6, so that the determinant b cannot be zero.
It follows that a '^ = e \ ; (H.6)
and therefore
that is,
and therefore (V?)' - - O 4 - 7 )
In this coordinate system the ground form at the point is
dx l *+...+dx 2 , (14.8)
and the first derivatives of (t ik ... vanish at the point. Of
course it is only at the point that these results hold.
15. Two symbols of four indices which are tensor
components. Let us now consider the expression
We see that since
_3 ~ M r MI M k M h Ciczf f
(15.1)
that is, the expression is a tensor component which should be
denoted by T r j c!iiy but as is customary we denote it by
(rkhi). (I 5 - 2 )
This is Christotfel's four-index symbol of the tirst kind.
We see that if the two first integers are interchanged the
sign is reversed, if the last two integers are interchanged
the sign is reversed, and if the two extreme integers are inter-
changed and also the two middle integers there is no change.
The expression a kt z ^ (15.3)
is -a vector whose components are tensgr components : it is an
associate vector to z. it and may be denoted by z k .
20 TENSOR THEORY
We then have z . ri z[ z. llL z(
= a li (rkhi). (15.4)
Tliis is Christoffel's four-index symbol of the second kind,
which should be denoted by J[ rhi > but is denoted by
{rthi}. (15.5)
Like the four-index symbol of the first kind it is a tensor
component. If the last two integers are reversed the sign
is changed, so that {rthi} = - {riih}. (15.6)
The three-index symbols, it will be remembered, unlike
the four-index symbols, are not tensor components.
We can express the four-index symbols in terms of the
fundamental tensor components ct ik ... and their derivatives.
We have
(rkhi) = -/A- = Xh-s-ik'
+ z rh z t ( ijct } >
and, as z^ = - (r/7), z^ = ~ (rht),
* *
V7 Z >'i Z l;~~ l\^ Z >h Z k - Z ii-]th z rh s 1ii>
\J tJL/1. ^+*^.^s vJUi >^ __ -^ s >^ ^ *^- ^/
ft ^~ i - P -
we therefore have
(rkhi) = ^- (rhk) - ^- (rile) + (rit) { kht } - (rht) {ikt}. (15.7)
cXi cx h
This formula may be written
(rkhi) = i(rhk)-h(rik),
if we make the convention that the operators are only to act
on the last integer, the first two being regarded as fixed,- and
the last as a lower integer.
We also have
{rkhi} = a kt (rthi)
= at 1 (I (rht) -h (rit))
= ia kt (rld)-ha ht (rit)
= i {rltk}~h {rik} ;
TWO SYMBOLS OF FOUR INDICES 21
and therefore
{rkhi} = ^- {rhk}- ^- \rik} + {tik} {rht}- [thk] {rit},
*** *** (15.8)
since the last integer in {ikt} is to be regarded as an upper
integer.
It may be noticed that
{ikt} (rht) = a*P (ikp) (rid) = (ikp) {rhp}, (15.9)
so that in the product {ikt} (rht) the two symbols { } and ( )
may be interchanged.
16. A four-index generator of tensor components from
tensor components. If we consider the expression
(3-9$ 7*; ?;;;;, ue.i)
we see at once that it is a tensor component. To find out
what it is we employ Euclidean coordinates at a specified
point.
At this point we see that it is
that is,
that is,
At the specified point we therefore have
w-> = {^>} ^)- {/^%>l (^) ; (iG . 2)
and, as this is a tensor identity, it must therefore hold at
every point.
The proof of this important theorem is a good example of
the utility of Euclidean coordinates, at a point. The three-
index symbols of Christoffel vanish at any point when referred
to the Euclidean coordinates of that point. If they had been
tensor components they would therefore have vanished in
22 TENSOK THEORY
any system of coordinates. The four-index symbols do not
vanish when referred to Euclidean coordinates. The four-
ind'ex symbols and the tensor components which are associate
to them are the indispensable tools of the calculus when we
apply it to differential geometry and to the Modern Einstein
Physics.
17. Systems of invariants. We have ( 12)
A (a) = a M^u A ,
and, in accordance with the notion of associate tensors, we
may write u k ^ a^ c u. ,
and therefore A (u) u^u^ (17 . 1)
Similarly we have
A (u, v) = ti f v t u t v f . (17.2)
In accordance with the same notion of associate tensors
we might say that u = u ik u . ^ ; (17.3)
but this is a rather dangerous use of the notation, as it
suggests that the u on the left is the same as the u from
which we formed u.^, which is absurd. However, a very
moderate degree of caution will enable us to use the Calculus
of Tensors without making absurd mistakes on the one hand,
or, on the other hand, introducing a number of extra symbols,
and thus destroying the simplicity of the calculus, for the
sake of avoiding mistakes which no one is likely to make.
We have proved, in 6, the formulae
-^- a* -f a u { tpq }+a&{tpi} = 0,
x p
a* = a* {pip},
t
and therefore we have
-- {pqi}.
*^t
It follows that ttla'*^ = a*a ik u . ih , (17. 3)
/
and therefore &2\ u ) a"*^ a*u l . (17 . 4)
SYSTEMS OF INVAKIANTS 23
If wo have any invariant of tbe quadratic form a^dx^dx^,
say 0, we can obtain other invariants A (0), A 2 (0) by means
of the differential parameters; and when we have two
invariants, <f> and \Jr, we also have the invariant A (0, \jr).
Clearly there cannot be more than n independent invariants.
Suppose that we have obtained, in any way, n independent
invariants tt 1 ,...u ri . Here the suffixes have no meaning of
differentiation or of being tensor components.
If \ve take the^e n invariants as the variables, then we have
a tk =A(v i u Jl ), (17.5)
and we can express the ground form in terms of the in-
variants.
In this case we can say that the necessary and sufficient
conditions that two ground forms may be equivalent that
is, transformable the one into the other are that for each
form the equations
A 0'i Uk) = 0toK---'M'fi) ( 17 - 6 )
may be the same.
For special forms of the ground form we may not be able
to find the required n invariants to apply this method. Thus
if the form is that of Euclidean space there are no invariants
which are functions of the variables.
IS. An Einstein space, and its vanishing invariants.
Let us write A rldh ~ ( rklh) , (18.1)
then { rkih} = <^A rpih , (18.2)
and therefore (rkil) ^ {rpih}.
We form associate tensor components (11) of (rkih)...,
and we know that they will be tensor components. Thus
we know that a M ( r j^ (18.3)
will be a ten or component. We write
A space for which all the tensor components A ij ... vanish
is .what is called an Einstein space. A space for which
24 TENSOR THEORY
where m is independent of the integers r, A, is called an
extended Einstein space.
We can form invariants from the associate tensor com-
ponents. Thus aik A ik (18- 4 )
is an invariant which we may denote by A.
Again, d k A ip (18.5)
is a tensor component which we may write A k p . We thus
have the series of invariants
A' P A*, A' P A*AI A' p A*A q r A r t . ... (18.6)
All of these invariants vanish for an Einstein space.
We can form another series of invariants which do not
vanish for an Einstein space. Thus we have
}>qra , (18.7)
and so on.
CHAPTER II*
THE GROUND FORM WHEN n = 2
19. Alternative notations. We now consider the ground
form a^dx^dx^ for the particular case when n = 2. That
is, we are to consider the geometry on a surface which lies in
ordinary three-dimensional Euclidean space.
The square of an element of length on any surface is
given by ds* = a u dx[ + 2a^dx Y dx^ + a^dxl t (19.1)
where o n , a 12 , L , 2 are functions of the coordinates x^ x 2
which define the position of a point on the surface.
We often avoid the use of the double suffix notation, and
take u and v to be the coordinates of a point on the surface,
when we write ds 2 = edu? ^ 2fdudv + gdv* ; (19.2)
or in yet another form
(fe a = A 2 du 2 + 2ABcosotdudv + B 2 dv 2 , (19.3)
where a is the angle at any point between the parametric
curves, that is, the u curve along which only u varies and
the v curve along which only v varies, and Adu and Bdv are
the elementary arcs on these curves.
There is no difficulty in passing from one notation to the
other. The double suffix is the one in which general theorems
are best stated : it alone falls in with the use of the tensor
calculus which so much lessens the labour of calculation.
* [The packets of MS. containing Chapters II and III, as submitted Jo
the Delegates of the University Press, were numbered by the author in the
reveiso order, and that order would probably have been made suitable, by
some reai rangement of matter, had he lived to put the work in readiness
for printing. It has seemed best, however, to revert to the order of a list
of headings found among the author's papers, an order in which the
chapters, as they stand, were almost certainly written.]
2843 E
26 THE GROUND FORM WHEN n = 2
In connexion with the form
ch 2 = edu 2 + 2fdudv + gdv 2
we use h to denote the positive square root of egf 2 : that is
h = a* = ^5sina, where a = a n a 22 a^. (19.4)
The element of area on the surface is
hdudv = ABsinadudv = a*dudv. (19.5)
2O. An example of applicable surfaces. If we are given
the equation of any surface in the Euclidean space, we can
express the Cartesian coordinates of any point on the surface
in terms of two parameters and thus obtain e,f, g in terms
of these parameters,
= &! +2/1+3?, /= i2 + 2/i2/ 2 + 2; i^ 9 = l+yl + *l>
(20 . 1)
where the suffixes indicate differentiation with regard to the
two parameters.
Thus, if u is the length of any arc of a plane curve, we
may write the equation of the curve y = <f> (u), and the
surface of revolution obtained by rotating the curve about
the axis of x will have the ground form
where v is the angle turned through.
Can we infer that, if a surface has this ground form, it is
a surface of revolution ? We shall see that we cannot make
this inference.
Thus consider the catenoid, that is, the surface obtained by
the revolution of the catenary about its directrix. The
ground form is fa* _. c j u z + ^2 + c ^ j v ^
Take the right helicoid, given by the equation
z =
x
this is clearly a ruled surface, and we can express the
coordinates of any point on it by
i
x = u cos v, y = u sin v, z = cv*
AN EXAMPLE OF APPLICABLE SUKFACES 27
Its ground form is then
and it is not a surface of revolution.
It is, however, applicable on the catenoid ; two surfaces
which have the same ground form being said to be^applicable,
the one on the other.
21. Spherical and pseudospherical surfaces. The
tractrix revolution surface. There are two distinct classes
of theorems about surfaces : there are the theorems which
are concerned with the surface regarded as a locus in space ;
and there are the theorems about the surface regarded as
a two-way space, and not as regards its position in a higher
space. It is the latter type of theorems about which the
ground form gives us all the information we require.
Thus all the formulae of spherical trigonometry can be,
as we shall see [in the next chapter], deduced from the
ground form <; 6 2 _ rfu 2 + S ; n 2 u ^ (21.1)
where u is the colatitude and v the longitude.
We shall prove the fundamental formula
cos c = cos a cos b + sin a sin b cos (7, (21 . 2)
and the formula for the area
4 + .B-K7-7T, (21.3)
and from these all the other formulae may be deduced.
So from the ground form
cL 2 = du 2 + sinh 2 udv* (21.4)
we can obtain the formulae of pseudospherical trigonometry
the trigonometry on a sphere of imaginary radius.
The fundamental formula is here
cosh c = cosh a cosh b sinh a sinh b cosh C, (21.5)
and the area of a triangle is
n-A-B-C. (21.6)
If in (21 . 4) we make the substitution (c being a constant)
u = u f c, v = 2e~ e v' 9
this ground form becomes
28 THE GROUND FORM WHEN n = 2
and if we take c to be a large constant it approximates to
the ground form ( / 8 2 _ du z + e -^dv 2 9 (21.7)
and to this form pseudospherical trigonometry will also apply.
The formulae of spherical trigonometry or of pseudo-
spherical trigonometry will apply to any surfaces which have
the same ground form as the sphere or the pseudosphere.
A real surface may have as its ground form
d** = du* + e-* u dv\
Thus if we take a tractrix, the involute, that is, of a
catenary which passes through its vertex, the equation of
the catenary is y _ cos h x , (21. 8)
taking the directrix of the catenary as the axis of x ; and if
we take u as the arc of the tractrix, measured from its cusp,
the vertex of the catenary, the equation of the tractrix is
y = e- u . (21.9)
If we now revolve the tractrix about the axis of x we get
a surface of revolution with the ground form
d6* = du* + e-* u dv*. (21 .7)
The figure of the tractrix is something like Fig. 1 ; and its
surface of revolution like Fig. 2.
22. Ruled and developable surfaces. The latter ap-
plicable on a plane. Let us now consider the most general
ruled surface, formed by taking any curve in space as base,
or as we shall say as directrix, and drawing, through each
point of the directrix, a straight line in any direction
determined by the position of the point on the directrix.
If x, y t z are the coordinates of any point on the directrix,
and I, m, n the direction cosines of the line, then [these
coordinates of the point and these direction cosines of the
line will be functions of a parameter v. We take u to bo
the distance of any point on the line from the point where
the line intersects the directrix. Then the current coordinates
of any point on the line may be written
a = x 4- ul t y' = y + um, z' = z + un ;
Fio. 1.
Fio. 2.
30 THE GROUND FORM WHEN H = 2
and for the ruled surface we have the ground form
d* 2 = du 2 + 2fdu dv + gdv 2 ,
where
(22.1)
' 2 + m^
(22.2)
That is, / is a function of v only and # is of the form
where a, /3, y are functions of v only.
We have ds 2 = (du +/cZy) 2 + (g -/ 2 ) cfe 2 ; (22.3)
the coordinates of any line of the ruled surface are functions
of v only, and therefore the shortest distance between the
point u y v and a neighbouring point on the line whose
coordinates are functions of v + dv is (gf 2 ) dv 2 .
The value of u for which this shortest, distance will be
least is then given by -^ = ; that is, the equation of the
oil/
line of strict! on is ^ n
V - = 0. (22.4)
dl6 V '
If we take, as we may, the directrix to be a curve crossing
the generators at right angles, and dv to be the angle between
two neighbouring generators, we have
ds 2 = du 2 + ((u - a) 2 + b 2 ) dv\
where a and b are functions of v. The line of striction is
now u = a, and the shortest distance between two neighbour-
ing generators is bdv.
For a developable surface therefore we have
d* 2 = du 2 + (u- a) 2 dv 2 . (22.5)
If we take
u' = u sin v la cos vdv f v' = u cos v | a sin vdv, (22 . 6)
we see that referred to the new coordinate system
(22,7)
so that the above transformation formulae establish a corre-
RULED AND DEVELOPABLE SURFACES 31
spondence, between the points on any developable and points
on a Euclidean plane, such that the distance between neigh-
bouring points on the developable and the distance between
the corresponding neighbouring points on the plane are the
same. The developable is therefore said to be applicable on
the plane.
23. Elliptic coordinates. Consider now the system of
confocal quadrics
992
* 2 , y _ ,_ _?*
a 2 -f u b 2 + u c 2 + '16
We know that the relations
2 _ (a 2 -f u) (a 2 + v) (a 2 + w) 2 __ (b 2 + u) (b 2 -f v) (b 2 4- w)
( ' '
give the coordinates of any point in space in terms of the
focal coordinates u, v, w ; and that the perpendiculars from
the centre on the threo confocals through any point are
given by
2 _ - 2 =
~~ ' ~~
(v u)(vw)
2 4- w) (b 2 4- w) (c* + w)
2 __
~~ w uw v
From the formula
p* = (a 2 + u) cos 2 a + (6 2 + u) cos 2 + (c 2 + u) cos 2 y, (23. 2)
where x cos a + y cos /? + z cos y = p
is the tangent plane to the surface u = constant, we see that
= du, 2qdq = dv, 2rdr = dw,
and therefore 4 cfe a = + + - (23 . 3)
p^ g 2 r^
If we now take w = and write
* U\ a 2 + v=F 8 , K{^^~b\ K\ = a 2 ~c 2 ,
32 THE GROUND FORM WHEN H = 2
so that the new coordinates are the semi-major axes of the
two confocals through any point on the ellipsoid
we Jmve
* y
*-K*J
K*)(U'-K*) (V*-Kl)(V*-K*)
(23.4
It follows, as a particular case, that the ground form for
a plane may be taken to be
,.,-, .^L. ,.
We thus have as ground forms of a plane
d* 2 = dx 2 + dy 2 ,
/ 2 2\
= (u 2 - v 2 ) ( -~ 5- -f -
'
and we could find an infinite number of other forms for the
plane, or for any other surface.
We are thus led to inquire as to the tests by which we can
decide whether two given ground forms are equivalent ; that
is whether by a change of the variables the one form can be
transformed into the other.
24. The invariant K. A0 and A 2 when K is constant.
Consider the form a ., ^ ,7^ _ t o
a ik ax i (lx k) ^ *> ^>
and let us use the methods of the tensor calculus.
In terms of the four-index symbols of Christoffel we have
one and only one invariant
(1212)-ra,* (24.1)
where a = a n ^ 22 ~ (^] 2 ) 2 '
* [The invariants A of (18. 4) reduce to one. Also, as the equalities
(1212) = -(2112) = -(1221) (2121)
hold, and the other symbols (1112), &c., vanish, the sum equal to (1212)' in
(15 . 1), with the notation of (15 . 2), is
For the explicit expressif n of K see Chap. Ill, 43.]
THE INVAIUANT K 33
We denote this invariant by K. ^
Let us first take the case when K is a constant and con-
sider the differential equations
(24.2)
We shall prove that they form a complete system : that is,
a system such that no equation of the first order can be
deduced from them by differentiation only.
We have . m + A' u & = 0, <f> . m + Ka^ = 0,
and therefore [ 16]
that is, ~{H 12] (f> f + K (a ll 2 - la 1 ) = 0. (24.3)
We have then to show that this is a mere identity.
Now {It 12] = a f l'(l2) 12) = u t2
and therefore
{1*12} fc =
so that the equation of the first order turns out to be a mere
identity. Similarly we see that the other equation of the
first order is a mere identity.
If and -v/r are any two integrals of the complete system
we have d
i (
p
- a * k (0 . ip + k + 0^ . Jq} ) + K
= 0. (24.4)
We therefore have
A (0) + K 2 = constant. (24.5)
We also have at once from (12.5) and the equations (24 , 2)
A 2 (0)+2jff0 = 0. (24/6)
34 THE GROUND FOKM WHEN U = 2
25. Determination of a \jr such that A (0, \/r) = 0. We
shall now prove that if we are given any function u, such
that A(u) and A 2 (u) are both functions of u t then, in all
cases (not merely when K is a constant), we can obtain by
quadrature a function v such that A (w, v) = 0.
Let J*W*u
H = e J A W , (25.1)
, , A 2 (u)
then *=-*. I
The condition that
where u 1 == a 11 u 1
may be a perfect differential is
that is, fi a* A 2 (u) -f a* (//j u 1 -f /^ 2 u 2 ) = ;
and this condition is fulfilled.
We can therefore by quadrature find a function v such that
v 1 = / ua^u 2 , Vgsr-^aht 1 , (25.2)
and therefore ^u 1 -f v 2 u 2 = 0,
that is, A(u, v) = 0. (25.3)
28. Beduction of a ground form when K is constant.
Returning now to the case when K is a constant, we have
seen that, if is an integral of the complete system,
A 2 (0)4 2#0 = 0, A ($) + /f< 2 = constant,
and we can therefore by quadrature obtain \fr t where
A(0,V) = 0.
First let us take the case when K is zero.
Without loss of generality we may suppose that
A(0)=l, A(0,^)=0, (26.1)
and we may take as new variables
x l = 0, aj a = ^r,
and the ground form becomes
cZs* = dx\ 4-tf 22 efce*.
REDUCTION OF A GROUND FORM WHEN K IS CONSTANT 35
Since x l is an integral of the complete system, we have
{111} = 0, {121} = 0, {221} = 0, .
and therefore (111) = 0, (121) = 0, (221) = 0.
From the fact that a 12 is zero, we have
(122) + (221) = o,
and therefore (212) = ;
so that a 22 is a function of OJ 2 only.
We can therefore take the ground form to be
cZs 2 = dx\+dx*. (26.2)
We next take the case when K is a positive constant, say
R-*. We then have
A (0) + jR~ 2 2 = constant,
and, without loss of generality, we may suppose
A(0) = ,R- 2 (l-0 2 ), (26. 3)
and, by quadrature, we can find ^ so that
A (0, f ) = 0. (26 . 4)
Take as new variables
x^ R cos' 1 0, # 2 - i//-, (26 . 5 )
and the ground form becomes
cZ& 2 = dx\ +a t22 dx%.
We have, since a 12 is zero,
(122) + (221) = 0,
/>
and, since cos-~ satisfies
= 0,
we have (221) + ^p c ot( J) = ;
and therefore (2 1 2) = cot
that is, ~a 22 = 2a 22
36 THE GROUND FORM WHEN U = 2
so that sin 2 ( fl)
C/ 22
is a function of rr 2 only.
We may therefore take the ground form as
(/6- 2 = dxi + sin 2 -,~ dx'] ,
or if we take ^ = Rx\, x\, = ^/ 2 5
we may take the ground form as
ds 2 = 2 (dU-J + sin 2 ^rficS). (26 . 6)
When K is a negative constant JK~ 2 , we see that the
ground form is ^ _ -2l 2 (dxl + sin 2 0^/0;:;) ; (26 . 7)
or if we take x\ LX V x' 2 a\ 2 ,
the ground form becomes
da* = Ji 2 (//^'J +sinh 2 a; 1 ^.]). , (26 . 8)
We have seen in 21 how the ground form
dtp = R* (dxi \-e~"**dx\) (26 . 9)
may be deduced from this.
27. The case of A (K) = 0. We have now seen the
forms to which the ground forms are reducible when the
invariant K is a constant; and we see that the necessary and
sufficient condition that two ground forms may be equivalent,
when for one of them K is a constant, is that for the other K
may be the same constant.
We must now consider how we are to proceed when K
is not a constant.
If A (K) is zero, we choose as our variables x l = K, x> 2 = v,
where v is any other function of the coordinates of the
assigned ground form.
Since A(a; 1 ) is zero, a 11 is zero and the ground form may
be written
dip = edu <2 + 2(f) 2 dudv y (27.1)
where e and </>., ( = -- ] are some functions of u and v.
THE CASE OF A (K) = 37
The equation which determines the invariant K which we
have taken to be u is therefore
or (2 02 __ 2u(/>) = 0. (27.2)
We may therefore take
e 2o
where a and /? are functions of u only.
The ground form now becomes, if we take
x l = K, x 2 == 0,
efe 2 - (ajjoj^ + otx 2 + /3) dx\ + 2 cfoy7.r 2 , ( 27 3 )
where a and /3 are functions of a; x only.
We can then decide at once whether two ground forms for
each of which A (K) is zero are equivalent.
28. The case when A 2 7i and AK are functions of K.
We may now dismiss this special case when A (K) is zero: it
is not of much interest, as it cannot arise in the case of
a real surface.
We now consider the case when K is not a constant and
A (A") is not zero, but A 2 (K) and A (K) are both functions
of /{". This arises when the surface is applicable on a surface
of revolution.
Let us take u = K t (28 . 1)
and let v be the function which we have seen can be obtained
by quadrature to satisfy the equation
A(u,v) = 0, (28.2)
when A 2 (u) and A (u) are both functions of u t though the
reasoning would have held equally had A 2 (u) ~- A (u) been
only assumed to bo a function of u.
We saw that if f A * (n \/ ?t
38 THE GROUND FORM WHEN H = 2
If then we take 6 = \idu^
we' have () l = pu^ 2 = /zi^.
and therefore ^ = a! (a 12 ^ + a 22 2 ) = a*0 2 .
Similarly wo have v 2 = alQ 1 ;
and therefore t = a^v 2 , 2 = alv 1 .
It follows that tf^ 1 = v 2 v 2 , 2 2 = 14 v 1 ,
and therefore A (fl) = A (v), A (0, v) = 0. (28 . 3)
We also have (c fl) 2 = (yucJu) 2 , A ((9) = (/z) 2 A (tt) s
and therefore ^ = ^- (28.4)
A((?) A(u) 7
If we now take as the new variables 6 and v, the ground
form becomes ^2 dv z
A ((9) '
We therefore see that the ground form may be written
fA ? (*
J A W
rf 2 = (A (A^)- 1 ((dKf + c J A W Jv 2 ), (28.5)
where v may be expressed by quadrature in terms of K and
integrals of functions of it.
We thus see that given two ground forms, for each of
which A (K) is a function of K and also A 2 (K) is a function
of K, the two forms are equivalent if, and only if, the functional
forms are the same.
29. Conditions for equivalence in the general case.
Finally we have the general and the simplest case when K is
not a constant, and A (K) and A 2 (K) are not both functions
of K.
In this case we have two invariants, say u and v. We
CONDITIONS FOR EQUIVALENCE IN THE GENERAL CASE[ 39
take these invariants as the coordinates, when the ground
form becomes
, 2 __ A (v) (fa, 2 - 2 A (u, v) dudv 4- A (u) dv*
~ * ' ( }
The necessary and sufficient conditions, that two such
ground forms may be equivalent, are that, for each of the
forms, A (w), A(u, 9 v), A (v) (29.2)
may be respectively the same functions of u and v.
We now know in all cases the tests which will determine
whether two assigned ground forms are, or are not, equivalent.
30. The functions called rotation functions. When the
measure of curvature * is constant we saw [ 24] that the re-
duction of the ground form to its canonical form depends
on finding an integral of the complete system of differential
equations
0. n + /fa u = 0, <f>. n + Ka n (f) = 0, (/> . 22 + Ka 22 (f> = 0.
(30.1)
We shall now show how this integral may be found by
aid of Riccati's equation.
Take any four functions, which we denote by q l9 y 2 , ? 1 19 r 2
and which will satisfy the three algebraic equations
Ka u = ql +rj, Ka u = qfa + r^, Ka^ = q$+r*. (30 . 2)
The functions thus chosen are not tensor components, but
we shall operate on them in accordance with our notation
with 1 and 2.
These two operators annihilate Ka n , Ka l2 > /ia 22 , and there-
fore we have
+ r l r 1>a = 0,
We define two other functions p l and p 2 by
* [This name for the invariant K will bo shown later to have geometrical
fitness. See 37.]
40 THE GROUND FO11M WHEN H = 2
It at once follows by simple algebra that
?V 2 + ?A,<7 2 ^= 0, </ 2 . 2 -j>V'2 =
We then havo i^ + ^iV/i +7V/1-2 = 0,
and therefore (21 12)?^ + q l (p } . 2 ?Vi) = 0,
that is, {1^12}^= <h(Pi-2-P*-i)> (30.4)
or a tk (1 i 12) r/ = ?1 (j> 1 .,-^ 2 . 1 ). (30 . 5)
Now
a tk (1 fc 12) r f = a' 2 (12 12) -^ =
and therefore
Similarly we luive
(30 - 8)
Aay six functions ^ , ^; 2 , ^ x , <7 2 , r ls r, 2 which satisfy these
three equations are called rotation functions. They havo
important geometrical properties and are much used by
Darboux, but here we simply regard them as algebraically
defined functions.
31. Integration of the complete system of 24. Now
consider the equations
A U
= 0, v -- mr 2 + 7ij a = ;
'
c' X'
/ 7
0, ^- - ^ a + Zr 2 = ;
INTEGRATION OF THE COMPLETE SYSTEM OF 24 41
These equations are consistent because the functions
Pi>Pv Vi,<l2> r i> r 2
are rotation functions ; and we see that I 2 -f- m 2 -f n 2 is a
constant.
Let q. = * + '_. , (31.2
V^ 2 + m a + 7t 2 -/J,
where stands for \/ 1 : wo have
We can therefore find or by the solution of Riccati's equation.
To determine Z, in, n we have
l + im , l im
cr = ""
and f 2 + m 2 + 7i 2 = constant :
and we thus see how, when we are given the rotation
functions, we can determine I, m, n.
We can now at once verify that
l. u + Ka n l = 0, I. l2 + Ka 12 l = 0, l. 2Z + Ka zz l = 0, (31.4)
and that A (1) = K (m* + n 2 ) = K (constant - 1 2 ). (32.5)
We have thus shown how a common integral of the com-
plete system can be obtained by aid of Riccati's equation.
The I, m, n which we thus obtain will be the direction
cosines of the normal to the surface, but we do not make any
use of our knowledge of a third dimension in obtaining
I, m, n.
CHAPTER III*
GEODESICS IN TWO-WAY SPACE
32. Differential equation of a geodesic. We have now
considered the ground form of a surface, and wo know the
method by which we are to determine when two given ground
forms are equivalent; that is when they are transformable
the one into the other by a change of the variables.
We now wish to consider the geometry on the surface
regarded as a two-way space ; and we are thus led to the
theory of geodesies. We have
ds 2 = UikdXidXfa (32.1)
and therefore
dSs __ dx A dSx k dx k dSx { cfo. dx k
2 7/iT "" ''* ds ^T + ik di ds ' + ds "57 ik '
d
<fai <lx k d,. fe
T ~T~ ; ' ~ : ^ - OX*.
ds ds dx t
For a path of critical length we therefore have
d / </./-,. \ d/ dx,.\ <*(/*,. dxjdx,.
~T ( a it ~j ) + -/-( a th i ) = -^ ,r-' (32 . 2)
ds\ H dd/ ds\ tlc ds / dz t ds ds v '
Now ^ = (/tt) + (*);
o^
and we notice that, though (itk) ^ (kti), yet
^ ^ fc __ a . dx 4 dx L
(Uk} 'ds lfc- (Ul >~ds'~ds (32 ' 3 ^
* [See foot-noto on p, 25.]
DIFFERENTIAL EQUATION OF A GEODESIC 43
The path of critical length therefore satisfies the equation
(/tfA. '(32:4)
'
33. Another form of the equation. The expressions
dx. , flx>,
-- 1 and ~
a 6- ds
are called the direction cosines of the element. They determine
the position of the clement at the point x ly # 2> but they are
not, in general, the cosines of the angles the element makes
with the parametric lines. We denote them by l , 2 in tensor
notation.
For a geodesic wo therefore have
We can put this equation in another form. We have
and therefore
"
it
Now gp (tpi) = ^^ (tki) = g k (kti)
and pg* (Ltk) = ggt (kti).
We then have for a geodesic
and therefore, multiplying by a!*! and summing,
For a geodesic we thus have either of the two equivalent
equations ,/
'^ = 0. (33.4)'
44 GEODESICS IN TWO-WAY SPACE
84. Condition that orthogonal trajectories be geodesies.
If two elements at x 19 x 2 are perpendicular to one another
we* have
a ll dx l Sx l -f a ]2 (dx v 8x^ + dx^Sx^ + a 22 dx 2 8x 2 = 0. (34 . 1)
The elements perpendicular to the curve = constant will
then satisfy the equation
2 (a n dx t + a l2 dx t2 ) (f> l (a^rf^ +tt 22 tZoJ 2 ), (34.2)
or "n^ + f^ 2 = M>
^ 1 + "22^ 2 = ^. ( 34 - 3 )
where /z is some multiplier.
We have
fi = ,1 (^0,-f a 12 2 ), ^ - ^ (^ l 1 + f( M a ), (34 .4)
and, as {&**= ]
we see that // f^ 1 X -f ^ 0J = 1 ;
and therefore /^ 2 A (0) = 1. , (34.5)
We thus have
L-, WW '+ M ^ = _ , (34.
(0) -/A (0)
7)
' ;
Now '~(a u g)-(i
ORTHOGONAL TRAJECTORIES 45
And, as A (<j>) = aPV<l> p <j> q ,
we see that (A (<j>)) t =
We therefore have
.
2A(0)~~ 2(A(0)) 2 >
Suppose now that is a function of the parameters such
tbat A (0) = F($). (34 . 9)
We see at once that the right-hand member of the above
equation vanishes; and therefore the orthogonal trajectories
of the curves <f> = constant are geodesies, if A (0) is a
function of 0.
Conversely we sec that, the orthogonal trajectories of any
system of geodesies being (f) = constant, A (0) must be a
function of 0.
35. Geodesic curvature. Jf l , 2 are the direction cosines
of an element of the curve constant, we have
ft? - . <'ikt't k = i,
and therefore
& = a*(A0)* 2 , 2 = -a*(A0)*'. (35.1)
Differentiating P<f> =
with respect to the arc, we have
that is, < / > (T;' > + {ikp} l j + Q'pq^^ = -
We therefore have, summing along the curve,
1 + { ^ 1 } tf^ds) ^ (d^ +
f.
+
(35.2)
The first integral if summed along a small length of the
curve only differs by a small quantity of the second order
46 GEODESICS IN TWO-WAY SPACE
from the same integral if summed along the curve formed by
the geodesic tangents at its extremities.
Now, summed along a geodesic, wo know that the first
integral vanishes, but summed along the curve formed by
two geodesies the integral is
IV-IV, (35.3)
where l , 2 and rj 1 , ?? 2 are tho direction cosines of the two
geodesies at their common point.
The angle a between the two elements whose direction
cosines are l , 2 and rj l , r? is given by
a*(V-V) = sina. (35.4)
We therefore have tho formula
a ~^ = (^^ )2 ~ 2 ^
(35.5)
where dO is tho small angle at which the geodesic tangents
at the extremities of ds intersect. The formula for the
geodesic curvature of the curve </> constant is therefore
^M0Mi(W a -20M20i& + 0. M (0,)>-Wr f . (35-6)
36. We can express the above formula in a better form :
to prove this we employ the coordinates which are Euclidean
at a specified point.
We have at tho specified point
and therefore
Now A 2 (0) = u + 22 ,
and therefore at the specified point
,A0, (36.2)
and at the specified point a is unity.
GEODESIC CURVATURE 47
We thus have at the specified point of the curve, and there-
fore at every point of the curve,
> (36 - 3)
The method of thus employing Euclidean coordinates is
very helpful in proving formulae in the tensor calculus. The
direct proof of the equality
(36.4)
would be much longer.
37. Polar geodesic coordinates. The measure of curva-
ture K. The geodesic curvature of a curve is given by the
formula
+ {ik2} *'
(37.1)
where g l t 2 are the direction cosines of an element of the curve.
If we take, and we shall see that we can take, the ground
form of the surface to be
du* + l?dv\ (37.2)
where u is the geodesic distance of any point on the surface
from a fixed point on the surface, and the curves v = constant
are geodesies passing through the fixed points, dv being the
angle at the point between two neighbouring geodesies, jB
being a function of u and v which on expansion in the neigh-
bourhood of the fixed point is of the form tt-f ..., where the
terms denoted by +... are of degree above the first, we
employ what we may call polar geodesic coordinates with
respect to the fixed point.
Let us now employ polar geodesic coordinates to interpret
the formula for geodesic curvature. We have
{111}=0, {112} = 0, {121} = 0, {122}=^,
{221} = -/?!, and {22f2}=jp. (37.3)
48 GEODESICS IN TWO-WAY SPACE
If d is the angle at which the curve crosses the geodesies
through the fixed point,
and [see 43] the measure of curvature K of the surface is
given by KB + B n = 0. (37.4)
We have
and therefore = -, h -W- sin 0. (37.5)
A/ ds B ^ '
Now consider the expression
(37.6)
where dS is an element of area of the surface, and take the
summation over the small strip bounded by two neighbouring
geodesies through the origin of the polar geodesic coordinates
and an element of the curve.
The expression is KBdudv,
and this is equal to B u dudv
r 7?
= dv- U^sintfds. (37.7)
rds
It follows that
Jpg
taken over the boundary of any closed curve surrounding
the point is equal to <. pp
P H (37.8)
MEASURE OF CURVATURE 49
where the integral is to be taken over the area of the
curve.
We thus have a geometrical interpretation of the measure
of curvature: it is the excess of 2?r over the angle turned
through by the geodesic tangent, as we describe a small closed
curve, divided by the area of the curve. It will be noticed
that in this definition we do not make use of any knowledge
of a space other than the two-way space of the surface itself.
This is what the curvature of a two-way space must mean
to a mathematician to whom the knowledge of a three-way
space can only be apprehended in the same vague wa} r as
we speak of a four-dimensional space.
38. Recapitulation. Parallel curves. It may be con-
venient to bring together the various formulae which so far
we have proved in connexion with direction cosines and
geodesies before we proceed further.
ds * ds '
where the direction cosines are those of an element of the
curve = constant ;
(34 '.7)'
where the direction cosines are those of an element perpen-
dicular to the curve ; for a geodesic we have
d ,
( I' / = 0- ( 33 *)'
The orthogonal trajectories of the curves (f> = constant,
where A (0) is a function of 0, are geodesies, and the ortho-
gonal trajectories of any system of geodesies are curves
(p = constant, where A (0) is a function of (f>. (34 . 9)'
Leaving aside the case when A (0) ^is zero, we can choose
the function so that A (0) = 1.
2843 II
50 GEODESICS IN TWO-WAY SPACE
If we know any integral of this partial differential equation
involving an arbitrary constant
<f>(x l9 x 2 , Oi) = 0,
then the system of curves /J
oft
will be geodesies : for A ( 0, ^~ J = 0,
and, as the condition that two families of curves
<p = constant, ty = constant,
may cut orthogonally is A (0, ty) = 0, we conclude that the
CUrV6S ^ = (38.1)
da
will be geodesies, since they cut the curves = constant
orthogonally.
If we choose the arbitrary constant /? so that the geodesies
given by ^0 ^
d a"' 3
may all pass through a fixed point, and if we take the
equation of the geodesies to be v constant and take v as
one of our parametric coordinates and to be the other
parametric coordinate u, we have
A(u) = 1, A(tt, v) = 0.
The ground form of the surface then takes the form
du* + B 2 dv*, (37.2)
and in the neighbourhood of the fixed point, through which
the geodesies pass, we may clearly take from elementary
geometry that jB = 7t + . . . .
We thus have what we called the polar geodesic coordinates.
We have d<j> = <f> l dx l -\-(f> 2 dx 2) and therefore, l , 2 being the
direction cosines of an element perpendicular to the curve
= constant, the length dn of the normal element is given by
d<f> = ~
or * = ,/A70). (38/2)
RECAPITULATION. PARALLEL CURVES 51
The curves which satisfy the equation
A ((/>) = 1 (38 . 3)
are called parallel curves. We see thus that two parallel
curves cut off equal intercepts on the geodesies which cut
them orthogonally.
If particles, constrained to lie on a smooth surface and
acted on by no forces but the normal reaction of the surface,
are projected at the same instant, with the same velocities
normal to any curve, they will at any other instant lie on
a parallel curve.
From the theory of partial differential equations we know
that any curve on the surface will have a series of curves
parallel to it, though the finding of them involves the solution
of the equation A (0) = 1.
The explicit forms of the differential equations
of a geodesic are
a l + {lll}xl+2 {121} 1 * 2 +{221}aj;j = 0,
a : 2 -f{112}a'i+2 {122} 0^2+ {222} a* = 0, (38.4)
where the dot denotes differentiation with respect to the arc.
If we write the variables as x and y and let
dy (Py
= =
we have y = xp t y
and the equation of the geodesic becomes
</- {221 }>'' + ({222} -2 { 121 })^ 2
+ (2 {122} _ {]!!})>+ {112} = 0. (38.5)
39. Notes regarding geodesic curvature. Now consider-
ing geodesic curvature, in the figure on p. 52 P and Q are two
neighbouring points on any curve, PT and TQ are the
geodesic tangents at P and Q, and QM is an element of arc
perpendicular to the geodesic tangent JPTM.
By definition the geodesic curvature of the given curve at
52 GEODESICS IN TWO-WAY SPACE
P is the limiting ratio of the angle QTM to the arc PQ as Q
approaches P. We therefore have
= 2 /|?f> ( 39 -o
Pg J- W
and thus have the analogue of Newton's measure of curvature
of a plane curve for a curve on the surface. It is the geodesic
curvature only that has a meaning when we conline our
attention to the two-way space on a surface.
We have the formula
V (36 . 3)
Pg VA(0) ' ^" " '
and we may apply it to find the geodesic curvature of the
curve all the points of which are at a constant geodesic
distance from the origin, in the polar geodesic coordinate
system. We have ds 2 du? + K*dv\
= u,
and therefore = r- log B.
Pg ^
The curvature will be constant if, and only if,
B=f(u)F(v),
that is, if the surface is applicable on one of revolution.
The curvature will then only bo - , as it would be in
i * u
a plane, it -, 9 72,972
1 ' ds 2 = du* + u*dv*,
that is, if the surface is applicable on a plane.
If we take, the case where K is positive unity and
da 2 = du 2 -f sin 2 u dv 2 ,
yve see that the geodesic curvature of a small circle is cot u.
If we take the form
NOTES REGARDING GEODESIC CURVATURE 53
which is applicable to the tractrix or any surface applicable
on it, wo see that the geodesic curvature of the curves
u == constant is minus unity.
40. The formula for the geodesic curvature may be written
(40 n
Let fji be an interating factor of
where 1 = tt n 0,+a 12 2 , 2 =
so that
and therefore , - (/za0 ] ) -f r-~7 (/zc^0 2 ) = 0.
O&'i O &'.,,
Now a*A 2 (0) - ^0', A (0, /i) = <pn t ,
and therefore /z A 2 (0) + A (0, /z) = 0,
that is, A. 2 (0) + A (0, log /z) = 0.
The formula for curvature may therefore be written
(4()
This is an equation to give the integrating factor. When
the integrating factor is known we can find the function -v/r
by quadrature ; and, as
A (0,^)= 0^=0, (40.3)
we have then the equation of the orthogonal trajectories of
the curves = constant.
In particular when the curves = constant are geodesies,
we may take ^ (A0)~i, (40 . 4)
and we thus see that the orthogonal trajectories of any
system of geodesies may be found by quadrature.
In eneral we have
and thus the formula for the geodesic curvature, of the curves
= constant may be written
(40.5)
Pg
where the curves ^ = constant are the orthogonal trajectories.
54 GEODESICS IN TWO-WAY SPACE
41. Integration of geodesic equations when K is con-
stant. We have obtained the differential equation of a geodesic
on any surface, but, in general, we cannot solve the equation
we have arrived at. Sometimes we can. Thus when the
measure of curvature is positive unity we may take the
ground form to be dt p _ du * + sin 2 ^ dv ^ ( 4 1 u } j
We then have as the equation of the geodesic
d6
that is, cos -,- + cot u sin 6 = 0,
au
or sin 6 sin u = constant.
XT A ^
JMow sin = smu-j-)
as
(ID
and therefore sin z u ,~ = sin a, (41.2)
where a is some constant.
We could have obtained this equation directly, as we easily
see, by the rules of the Calculuy of Variations.
We deduce that
and therefore cos u = cos a cos s,
and we thus obtain the equations
sin 8 sin a cos s . tans . .
sm v = - , cos v = . ------- , tan v = -.- (4 1 . 3)
sin u sm u sin a
We now see that
cos 14 cos u 2 -f sin Uj sin u 2 cos (v l v 2 )
= cos 2 a cos Sj cos 8 2 -f sin 2 a cos s x cos s 2 -f sin ! sin s 2 ,
= COS (^ Sg).
This is just the well-known formula of spherical trigonometry
cos c = cos a cos 6 + sin a sin 6 cos (7. (41 . 4)
Similarly we could obtain the formula
cosh fo f 2 ) = cosh *j cosh u 2 sinh ^ t sinh u 2 cos (i^ v 2 )/
(41.5)
INTEGRATION OF GEODESIC EQUATIONS 55
which would be applicable to a surface of constant negative
curvature.
The formula
when applied to a geodesic triangle on a surface of curvature
positive unity, gives us the well-known formula for the area
of a spherical triangle A+B + CTT (41.6)
and more generally for any surface of constant curvature
TT). (41 . 7)
42. Focal coordinates. If we take, as the coordinates of
a point on a surface, tho geodesic distances of the point from
two fixed points on the surface, the ground form will take
the form ( s i n a )-a (^2 + dv 2 __ 2 cos a dudv),
where a is tho angle between the two geodesic distances.
We easily see this geometrically, using the property that
the locus of a point at a constant geodesic distance from
a fixed point is a curve cutting the geodesic radii vectores
orthogonally. Analytically we prove the formula from the
fact that A (u) and A (v) are both unity, and applying this to
the general ground form
A 2 ( In 2 + B 2 dv 2
when we have A 2 = B 2 = cosec 2 a.
If we take 2x
we have db 2 = sec 2 -- dx 2 -f cosec 2 - dy 2 . (42.1)
2 6
This system of coordinates may be called focal coordinates :
the curves x = constant will represent confocal ellipses ; that
is, curves the sum of whose geodesic distances from two fixed
points, which we call the foci, is constant.
Similarly the curves y = constant will represent confocal
hyperbolas, and we see that the ellipses and hyperbolas
intersect orthogonally.
.43. Explicit expressions for symbols {ikj} and for K.
It will be convenient here to give in explicit form Christoffers
56 GEODESICS IN TWO-WAY SPACE
three-index symbols of the second kind,* as we so often need
them, and expressions for the measure of curvature.
"We take the ground form
da 2 = cdu 2 -f 2/dudv + gdv*, (43.1)
and we then have a == It 2 egf 2 , (43 . 2)
* 2 -2/ 1 ), 2/^(112} =
=tf(2/ 2 -fl 1 )-/ S r 2 , 2 /, 2 {222 } = */ 2
(43.3)... (43.8)
4 A 4 # = e (r/ a ( 2 - 2/J -f gl) + flf (^ (^ - 2/ 2 ) + e )
If we take as the ground form
the last formula becomes
M n . ^ /R ul () cosa\ ^ /An B, cosa\
A B sin a A + a, 2 -f ^-- ( . .- -- ) + ^ ( Si~ ' - 1 = 0,
1Z CU\ ^i Bin OK / CV \ ,0.3111 OK /
(43.10)
which is Darboux's form.
* [Those of tho first kind arc at once
(HD-i^, (112) -^-ic
(121) (211) = J Ca , (122) (212)
(221) -/.H^, (222) = J^ 2 .
Wo also have
-h flf ( -
Cr ^-- f <>
2
EXPLICIT EXPKESSIONS FOB SYMBOLS 57
In particular if the parametric coordinates are geodesies
we have {221} = 0, {112} = 0, (43. ;i)
and therefore
rt A 9 ~B, cos a
2 A sin a ' x B sin a '
and the formula for the curvature takes the simple form
ABsin aK = 12 . (43 . 12)
From this formula wo could easily deduce again the formula
r*uffA'ci=2,r.
J p^ JJ
When we take the ground form to be
2
we have
When we take the ground form to be
we have
{111}=, {112} -> {121} =-' 2 -, [122}=*
1 j 2e ( 2e ^ 2e *- * 2e
(221 } = ~, {222} = - 2 -,
2e 2e
(43. 14) ...(43.19)
; 0, (43.20)
(43.21)
+ (!#)> (43.22)
Finally, when we take the ground form to bo
2fdudv,
58 GEODESICS IN TWO-WAY SPACE
^ 44. Liouville's special form. When the ground form of
a surface takes the special fdfm Liouville's form
17 and F denoting functions of u and v respectively, we
can find a first integral of the equation of geodesic lines.
For the form e
ci l = cos 0, ei 2 = wind,
and the equation of a geodesic becomes
^0 . A
2 e* + e : sin - 0o cos Q ~ 0,
that i, - (cl sin ^) + v (& cos 5) = 0* (44.2)
c l(j v V
We therefore have
e^ sin 6 = 0o, c^ cos = 0j ,
and e = 0i -f </>;;,
that is, A(0) = 1. (44. 3)
In the particular case of Liouvillc's surface
<t>i-U=V-M,
and \ve obtain a complete integral of this partial differential
equation by equating the above expressions to a constant.
We thus have ^ = ^(j~+a t <P, = SV~-a,
giving the first integral
e^cos^ = VU+a y 6*siii tf = -/F a, (44.4)
<^ 2 '^ 8 /^, r,
or y 7 - = TT -- (44.5)
45. Null lines. Complex functions of position. We
shall now consider a further application of Beltrami's differ-
ential parameters to the geometry of surfaces.
NULL LINES. COMPLEX FUNCTIONS OF POSITION 59
The null lines of a surface arc the lines which satisfy the
equation a ik dx^lx k = 0. (45 . 1)
These lines play the part in the geometry of a surface
which the circular lines play in plane Euclidean geometry.
If the equation of the null lines is <f> = constant,
then A (0) = 0. (45 . 2)
To obtain the null lines wo must therefore be able to solve
this equation.
The equation will have two independent integrals. If we
take these integrals as the parameters we employ what we call
null coordinates. The ground form takes the form
2fdudv, (45.3)
j i, x /*\ 2 ^0 ^0
and, as we have seen, A (<p) = - --- --- -
If then A (0) = 0,
must be a function of u only, or a function of v only.
A function satisfying the equation may bo called a complex
function of position. There are therefore only two types of
complex functions of position, viz. the two functions whose
differentials are multiples of the factors of d&. The first we
shall take as that which corresponds to the factor
(a n dx l + (a 12 + i Va) dx^ -f a n , (45 . 4)
and the second that which corresponds to
(a ll rfo; 1 -f ( 12 ' Va) dx^ -f- a n i. (45 . 5)
We need only consider those which correspond to the first
factor, and, if we do this, we can say that every function of
position is a function of every other such complex function.
Thus in the case of the plane, where we have
rfs 2 = dx z + dy* = dr* + r*dO*,
x + LIJ is a complex function of position since its differential is
a multiple (unity) of dx+idy of the first factor of dx^ + dif,
and log r + 1 d is a complex function of position since its
differential + idd is a multiple (-) of dr+ird0 of the
first factor of cZr 2 + r 2 d! 2 ; and log r -hid is a function of
60 GEODESICS IN TWO-WAY SPACE
Just as the position of any point in the plane is given by
means of the complex variable x + iy, so the position of any
point oh a surface is given by means of the complex variable
u where u is an integral of
A(0) = 0. (45.2)
46. Conjugate Harmonic Functions. Mapping on a
plane. Let,tts now consider the equation
A, (0)^0, (46.1)
that is, -~i0 l + 7 td(f>* = 0, (46.2)
C Ct'^ C) jC'n
where 1 = a u fa + a l *fa, 2 = a l2 fa + a**fa. (46 . 3)
The expression a! ^fdx l v& fa dx 2
is thus a perfect differential if A 2 (0) is zero; and we have
alp = fa, a*<l> 2 = -fa t (46.4)
and therefore cdty 1 fa, a*\p fa, (46.5)
It follows that A 2 (fa = 0, A (fa fa = 0. (46 . 6)
Thus if <f> is any integral of A 2 (0) = we can by quad-
rature find *//, another integral of the equation, and the two
curves = constant, ^ = constant will cut orthogonally.
A real function, annihilated by the linear operator A 2 of
the second order, is said to be a harmonic function. The
function >//-, obtained as explained by quadrature from 0, is
called the conjugate harmonic function to 0. It will be
noticed that the function conjugate to i/r is not <f> but </>.
We also have A (0) = A (i/r), (46 . 7)
and therefore, since A ((/>, \fr) = 0,
we see that A (0 + i\fs) = 0. (46 . 8)
The function + i^ is thus a complex function of position
on the surface.
If we take u = 0, v = y\r
we have cfo 2 = (A (0))- 1 (du* + dv 2 ). (46 . 9)
Thus the problem o/ mapping any surface on a plane, .so
that the map may be a true representation of the surface as
CONJUGATE HARMONIC FUNCTIONS .61
regards similarity of small figures in each, just depends on
the solution of the equation
M0) = 0. ' (46 '.!)
The magnifying factor from the surface to the plane
is A (<f>).
Thus to map any surface, applicable on a sphere of unit
radius, and whose ground form may therefore be taken as
du 2 4-sin 2 ucfe 2 , (46.10)
upon a plane we have A 2 ($) = ;
and this tells us that <f> must be a function of
, /. u\
logftan-J + iv.
\ t /
We thus obtain Mercator's Projection
(ni ^
tan-), y-v. (46.11)
The theory of conjugate functions of position on a surface
can be applied to problems in Hydrodynamics and Electricity
as has been done in the case of the plane. Thus if is
a harmonic function on the surface, we may take it to be the
velocity potential in the irrotational motion of a liquid over
the surface, and \/r, the conjugate harmonic function, will then
be the stream function.
Conversely, if the ground form is taken to be
ds 2 = e(du 2 + dv*), (46. 12)
u and v will be conjugate harmonic functions.
CHAPTEE IV
TWO-WAY SPACE AS A LOCUS IN
EUCLIDEAN SPACE
47. A quaternion notation. So far we havo been think-
ing of the two-way space associated with the ground form
ds* = a ik dx { dx lt ;
we must now think of that space as a surface locus in
Euclidean space.
Let i', t", t" r be three symbols which are to obey the
associative law and the following self-consistent laws :
, <Y" = -4",
iY = - 1, i'Y' = - 1, "Y" = - 1. (47 . 1)
Let a/, x" , x 1 " be three ordinary numbers called scalar
quantities, then, if x = x' i' + JL" i" + x'" i", (47.2)
x may be said to be a complex number.
If we take y = y ' L ' + y " L " + y '" /",
we see that
xy = - (x'y' + x"y" + x'"y'") + (x"y'" - x'" y") i'
+ (x"'y'-x'y'") i" + (x'y" -x"y') i"', (47 . 3)
so that xy consists of two parts, a scalar part and a complex
number. We write the scalar part
Say or xy, (47 .4)
and the complex part Vxy or xy. (47 . 5)
It follows that x 2 is a pure scalar.
We may easily verify the following results :
xy~yx = 2xy, (x + y) 2 = x 2 + y
A QUATERNION NOTATION
and, by multiplying the two matrices
63
y" 11
W' 10"
we verify that
tixyzw xwyz xzyiv,
Vxyz = zxy yzx,
Vxyz -f Vyzx -\-Vzxy-Q.
If we take i, i" y i" to be unit vectors in the positive
directions along the axes of rectangular Cartesian coordinates,
then x will be the vector from the origin to the point whose
coordinates are x', x", x"' . The length of the vector x will
be denoted by | x |. The symbol xy will denote a vector at
right angles to x and y, and in the sense that, if the left hand
is along x and the right hand along y, then the direction xy
will be from foot to head ; the magnitude of the vector will
be \x\ | y| sin 6, where is the angle between x and y from
left to right.
The scalar xy will be equal to | x \y\ cos 6.
48. Introduction of new fundamental magnitudes and
equations. Now let z be a vector whose components z\ z" , z"'
are functions of the parameters x l and o? 2 , that is, of the
coordinates of the two-way space. We have
dz = z p dx p
and
The vector z traces out a surface. Let the unit vector
drawn at the extremity of z normal to this surface be denoted
by A. We have proved [in 13] that z.^ is parallel to A.
We therefore have 3 .^ = /2^A,
where fl^ is a scalar quantity.
We know that
an.d therefore .
(48.1)
(48.2)
64 TWOWAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
Multiplying across by X and taking the scalar product
we have, since \\ h - , XX 7 , = 0, \z t = 0,
the equation fl ik . h = fl ih . % . (48.3)
This is true for all values of i, h, k from 1 to 2 inclusive,
and flfa = /2fc$, so that
-On-2 = ^i2-i ^ 2 . 1= /2 12 . 2 . (48.4)
These equations are known as Codazzi's equations.
49. Connexion of the magnitudes with curvature. The
length of the perpendicular from a point at the extremity of
the vector z + 8z (where 8z is not necessarily small) on the
tangent plane at the extremity of z is
-\8z. (49 -1)
If we now take Sz so small that cubes of 8x^ 8x% may be
neglected, the length becomes
that is, iC^n^i + 2f2 12 8x 1 8x 2 + fi, 2(2 $ x *)' (49 . 2)
The radius of curvature of any normal section of the
surface is therefore given by
* ^*^i^ (49. 3)
r\ il fl * ill' * '
j-i/ He, *^t /
in the tensor notation, and the principal radii of curvature
are consequently given by
l "-"' ^~^ 2 =0. (49.4)
The product of the reciprocals of the principal radii of
curvature is therefore /2 n /2 22 -/2f 2 (49 g)
Now we saw that
(1212) = ^.^^.^-^-ii -^'
= /2 n /2 22 -/2; a , (49 ..6)
and therefore the invariant K is just the measure of curvature.
CONNEXION OF THE MAGNITUDES WITH CURVATURE 05
We thus have the equations
Ka = n n nn-ni*> .(49-7)
n a . t = n u . lt A w -i-=A 22 > (48.4)
wherewith wo are to determine the functions
n u ,n M ,n M * (49.8)
When wo have found these functions we can find the principal
radii of curvature by aid of the equation
/2 w -/2J a = 0,
(49.9)
which may be written
i - ^n^+a-^n^^-n-,) = o, (49. 10)
applying the tensor notation to the coefficient of -^
If wo were to keep strictly to the tensor notation we should
write /2 n ,f2 22 /2| a as /2. We must distinguish between the
integer which denotes merely a power, as in /2j 2 denoting
the square of /2 12 , and the integer which we called the upper
integer in a tensor component. The two meanings are not
likely to cause any practical difficulty in reality.
SO. The normal vector determinate when the functions
f2jj c are known. We must now show how we may determine
the unit vector X when the functions n ik are known.
* [It is usual to speak of the functions fl u , n 12 , n 22 , i.e. (by 50) ZjAj,
SjAj = -2^i> ~a^2 as tho fundamental magnitudes of the second order, those
of the first order being the a u , a n , a.,^ or c, /, g of the ground form ds* y and
to say that tho six are connected by Gauss's equation (49 . 7), in which K
( 43) is a known function of the magnitudes of tho first order and their
derivatives, and by the two Codazzi equations (48.4). Written at greater
length these two equations are
sa -{i2i}n la = - n ja -{222}n ja -{22i}n ll ,
v x^ f- %i
and their explicit forms are obtained by substituting in these for {HI}, &c.,
from 43.]
2843 K
66 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
We denote the ground form of the spherical image, that is,
of the sphere traced out by a unit vector drawn through the
origin, parallel to the normal at the extremity of z, by
a'ij.tL'idXk, (50.1)
so that 0,'^ = A t -A 7r (50 . 2) ,
If X . ik =\ ik -{ikt}'X t ,
where {ilct}' refers to the ground form of the spherical image,
we see as before that X. iJc is parallel to the normal to the
sphere at the extremity of A : that is \. ik is parallel to A.
Now AA t - is zero, and differentiating we have
M ik + \i\ k = >
so that A A . ik + Ay A/. = 0.
It follows that \. ik = A;A 7( .A = -u' tV .A ; (50 . 3)
and as we have shown [in 30] how, when ' t -/ f ... are given, A
can be obtained by aid of Biccati's equation, we have only to
show how a' ik ... can bo expressed in terms of a ik ... and
Along a line of curvature we have
cfe+RdA = 0; (50.4)
let R f and R" be the principal radii of curvature, and let us
choose the lines of curvature so that they may be the para-
metric lines, that corresponding to R' being
efajj = 0,
and that corresponding to R" being
da^ = 0.
We therefore have
^H-jR'Aj = 0, z 2 + R"\ 2 = 0. (50. 5)
And it follows that
a n = R'f2 n , c* 12 = 7J'/2 12 , a 12 =ft"/2 I2l a 22 = /T/2 22 ,
/2 U = fi'a' n , /2 12 =.J2'a' M , /2 12 = JZ"ce' 12 , /2 W = E"a' i2f
(50.6)
THE NORMAL VECTOR TO THE TWO-WAY SPACE 67
so that a n -(J8 / + J B // )/2 11 + a / 11 B'JfZ / '= 0,
fl n + a r Vi ll'R" = 0,
tln + a'nR'&" = 0. (50.7)
Now the expressions on the left in these equations are
tensor components, and therefore, as they vanish for one
particular coordinate system, they vanish for all systems.
That is, the equations are identities.
We may express the identity in the form
dz* + (K + R") dzd\ + R'R"d\* = 0. (50 . 8)
We thus see how o/^... are obtained.
We see that n ik = ^ = z^ (50 . 9)
for \z i = 0,
and therefore ^ z ik + ^u z i >
which gives ^ z 'ik + ^k z i 0-
From the equations
Aj^ = /2 n , Ajj?,, = ^ = /2 12 , Ajjj 2 = /2 22 , A^j = 0, Xz^ =
(50.10)
we can find Sj and 2 when A is known, and thus determine z
by quadrature.
We have now shown how the determination of the surfaces
applicable to the ground form
depends on the determination of the functions fl^ .
But here comes the difficulty : the equations to determine
these functions
flu-2 = ^la-n Au-i = ^12 2' K* ^ fliA-flfa
are differential equations of the second order which, in general,
we cannot solve.
In one very special case we can solve them, viz. when the
invariant K is zero. In this case we have shown that the
ground form may be taken to be
(50.11)
68 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
The equations now become
- - u-wio -~ C u*i i , ~ "*|o "~~ C " *'2'2
^ 12 c^ 2 U ZX> 2 ' 2 ^ 22
and therefore
,
where
- (50 12)
2 ~ (J - ;
We can easily prove that we are now led to developable
surfaces.
51. Eeference to lines of curvature. The measure of
curvature. When we refer to linos of curvature as para-
metric lines we have, in (50 . 6),
a la = ,R'/2 12 , a 12 = J2"/2 12>
and therefore, unless R' and R" are equal, we must have
a la = .0,2 = 0. ^ < 51 - 1 .)
If the radii of curvature are equal, operating with I and 2
which annihilate a ik , we have
2 . 2 = 0. (51.2)
Similarly by operating on
a n = -R/2 U , 22 = J?/2 22 ,
we have li 2 if2 11 + JR/2 11 . 2 = 0,
JS 1 /2 22 + ^^22-1 = - ( 51 3 )
From Codazzi's equations we deduce that
As wo cannot have /2 n /2 22 /2 25
unless 7J is infinite, we must have
jRj = 0, J2 2 = 0, (51 .4)
that is, U is constant and the surface must bo a sphere.
Leaving aside the special case of a sphere, we have when
the parametric lines are the lines of curvature
a 12 = /2 12 = a' 12 = 0, (51 .5)
REFERENCE TO LINES OF CURVATURE 69
and wo can often simplify proofs of theorems by referring to
lines of curvature as parametric lines.
The vector z^z 2 is clearly normal to the surface at the
extremity of z : its magnitude is ai (or h as it is generally
written) and therefore z^ 2 = a^A.
Similarly we have ^1^2 ^ tt '^
The expression Kz i z k ~X i \j c
is a tensor component. It obviously vanishes when we refer
to lines of curvature: it therefore vanishes identically and
WehaVO K^z 2 = \X 2 . (51.6)
We then have Kal = a'*, (51 . 7)
that is, the measure of curvature is the ratio of a small element
of area on the spherical image to the corresponding area on
the surface.
62. Tangential equations. Minimal surfaces. We shall
now develop some further formulae. We have
/2/2,-/2? 2 =aK = a'K~ l = (aa';
n* = + jn. (52.1)
and, from the formulae connecting
a ik> u 'ik> ^ihi
we easily deduce a' ik fl ik = R' + R",
We can also obtain formulae applicable to a surface given
by its tangential equation. This means that instead of
beginning with a vector z, given in terms of parameters x l
and aJ 2 , we begin with assuming that X is known in terms
of these parameters, and also p, the perpendicular from the
origin on the tangent plane to the surface.
The lines of curvature are given by
= 0,
= 0.
70 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
They are therefore also given by
(n, n -Ra\ l )dx l -}-(n^~Rii' v ^dx^ = 0,
(/2 ia - JRu' 12 ) c/^-f (/2 22 - JKa' 2a ) tte a = 0, (52 . 3)
as we see at once from the connecting equations.
The tangential equation of a surface is
p + Xz = 0. (52.4)
By differentiation wo deduce that
With reference to the ground form of the spherical image we
therefore have p . ^ + x . ik z + f2 ih = 0.
Now A.^ = -a'. 7 ,A,
and therefore />. ^ + a'^> -f- fl^. = 0. ^ (52.5)
When therefore we are given the tangential equation of
a surface, the lines of curvature and the radii of curvature
are given by the formulae
i = 0,
(52.6)
In particular if we want the parametric lines to be lines of
curvature on the surface we must have
and therefore > must satisfy the equation.
*>.,= 0. (52.7)
There is a particular type of surface with which we
shall have to do : the minimal surface characterized by the
property that the principal radii of curvature are equal and
opposite.
The expression SAA^-SAA^ (52 . 8)
is a tensor component.. If the surface is a minimal one .it
vanishes when we refer to lines of curvature, and therefore
TANGENTIAL EQUATIONS. MINIMAL SURFACES 71
it vanishes always if, and, we see, only if, the surface is
a minimal one.
We always have the formula, as we easily see,
The tangential equation of a minimal surface is therefore
given by A'.p + 2^> = 0. (52 . 9)
If we refer to the null linos of the spherical image as para-
metric lines, the ground form of the sphere becomes
4 (1 + x l (i' 2 )- 2 dx l dx. t i
and the equation which p has to satisfy becomes
It may be shown by Laplace's method that the most general
solution of this equation is
(l+x^p = 2x l f(x 1 )+2,*\ 2 <f) (x 2 )
+ (1 +*- 1 ag (x\f (x^ + xl f (0 2 )), (52 . 10)
and we have thus obtained the tangential equation of the
minimal surface.
63. Weingarten or W surfaces. We now proceed to
consider more generally surfaces which, like the minimal
surface, are characterized by the property that their radii of
curvature are functionally connected. These surfaces are
called W surfaces, after Weingarten, who studied their pro-
perties.
When we refer to the lines of curvature as parametric lines
we have (50. 5) ^ + 1^ = 0, c 2 + #"A 2 ^0,
and therefore (R'-R") A 12 = JK''^- JR'.^.
Let R"=f(R r )
n dx
and (x) = ej *-/(*) .
We easily verify that
f (x) $ (x} <f>"(x) 1. 0'fa)
' - ' { '
72 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
Now
. R'.\ _ /' (K) R\ _ 0" (R>) R\ a
.'-." ~ R r -t(R'} ~ "~0'TT~ ~ ^ g '^ ( ^'
*' - JT - If -
The equation satisfied by A thus becomes
X v=\ 4 lo o' (*' W) - \ ^ log (^ (JZO), (53.2)
and therefore, since A,X 2 is zero,
We may therefore, the lines of curvature still remaining
the parametric lines, take
+l=0. (53.3)
The spherical image of the TT surface (that is, it will bo
remembered, the surface traced out by a unit vector parallel
to the normal at the extremity of the vector z, and expressed
in the coordinates which give z), when the W surface is
referred to the lines of curvature as parametric lines, will be
therefore 2
It will be sometimes more convenient to express the para-
metric coordinates by u and v.
Conversely, if we are given the ground form of a sphere
in the form pdu? + qdv*, where p and q are functionally con-
nected, it will be the spherical image of a W surface referred
to its lines of curvature.
54. An example of W surfaces. We may now consider
some examples. We saw ( 42) that, referred to what we
called focal coordinates, the ground form of any surface may
be taken as J 8 2 _ Sec 2 $ dv * + coscc a 0^ (&*.l)
where 2u = PA + PB, 2v = PA- PB,
and A and B are any two points on the surface which we call
the foci ; PA and P$ are geodesic distances and 2 6 is the
angle APB.
AN EXAMPLE OF W SURFACES 73
If the surface is applicable on a sphere we see that
. sin(c v)sin (c + v)
tan C7 = ".- , ------ r : ---- :
c)
where 2e is the geodesic distance between A and B.
Thus (#') = cos 6, <$>' (R') = cosec (9,
and therefore cosec 6 ^ = sin d.
civ
If we now integrate this equation we- have
R' = sin 2 0-20 4- e,
where e is some constant.
But JK'-jR" = ^S- = sin 6 cos (9,
(/> (R)
so that 4E" = -20- sin 20 + e,
and therefore 2 (R'-R") = sin (e - 2 E' - 2 .K"). (54 . 2)
This is the relation between the principal radii of curvature
of the W surface which corresponds to the spherical image
sec 2 6du* + cosec 2 6dv 2 . (54 . 1)
In this case we know the radii of curvature in terms of the
parameters since 6 is so known. We thus know the ground
form both of the surface and of the spherical image, and there-
fore can find the surface as a locus in space.
55. The spherical and pseudo-spherical examples. In
the above example we began with a known ground form for
the spherical image and deduced the relation between the
curvatures.
If we take any knowft ground form for the spherical image
2*lu 2 -f qdv 2 ,
where p and q are functionally related, and known in terms
of the parameters, we could proceed similarly. We could
find the relation between the curvatures and we should
obtain in known terms of the parameters the ground form of
the. surface. We could then obtain the surface as a locus
in space. In my exposition of the method I have followed
74 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
Darboux and taken the example he gives, as I #lso do in
what follows.
When on the other hand we begin with a known relation
between the curvatures, we cannot in general find the surface
as a locus in space. Thus, let us &pply the method to the
problem of finding the surfaces applicable on a sphere of
unit radius.
Here we have R' R" = 1, (55. 1)
and we may take R' = co th 0, R' 1 = tanh 6.
The function which expresses R" in terms of R' is
p dR'
PT T
and (f) (R') = e R' = cosech 0,
</>' (R f ) = cosh 0.
The ground form of the spherical imago is'thus
sinh 2 (9 c/u 2 + cosh 2 6dv*. (55 . 2)
On the sphere the measure of curvature is unity, and therefore
our formula for K gives
6 n + #22 + inh cosh 6 = 0. (55 . 3)
Now if we knew how to solve this equation we should
have an expression for in terms of the parameters u and v,
and we should thus be able to write down the ground forms
of the surface and of the spherical image in terms of the
parameters ; and thus have the means of determining as loci
in space all the surfaces which are applicable on the sphere.
Unfortunately wo cannot solve the equation generally.
This example shows how ultimately nearly all questions in
Differential Geometry come to getting a differential equation ;
and that the complete answer depends on the solution of the
equation. But even when wo cannot solve the equation we
gain in knowledge by having the differential equation in
explicit form. Thus it happens sometimes that two apparently
quite different geometrical problems may depend on the sqme
insoluble differential equation. The surfaces connected with
THE SPHERICAL AND PSEUDO -SPHERICAL EXAMPLES 75
the problems are thus brought into relationship with one
another; and the relationship ia sometimes very simple and
very beautiful. Illustrations of this will occur later. All we
can say now is that the differential equation
n 4- 22 + sinh 6 cosh =
is that on which depends the obtaining of all surfaces which
are applicable on the sphere : that is, the surfaces whose
geodesic geometry may be considered as absolutely known,
being just spherical trigonometry.
Similarly we might consider the problem of finding the
surfaces applicable on a pseudosphere. Here we have
JZ'li" = -l, (55.4)
and wo take R' = cot 0, R" - tan 0.
We find that
<f> (cot 0) coscc 0, 0' (cot 6) cos 0,
so that the ground form of the spherical image is
sin 2 0Ju 2 + cos 2 0(fo 2 , (55.5)
and the equation to determine 6 is
22 -0 u -fsin0cos0 = 0. (55.6)
If we apply the substitution
2ti'=w + v, 2v' = u-r, 20 = 0',
the equation takes the simpler form
12 = sin0; (55.7)
and on this equation depends the obtaining of the surfaces with
the known pseudospherical trigonometry, obtainable from
spherical trigonometry by writing ia, ib s ic, for the arcs of
a spherical triangle.
66. Keference to asymptotic lines. We have now con-
sidered the surface when referred to lines of curvature as
parametric coordinates, and the equations resulting,
z l = U'Aj, 2 = R"\>
where R' and R" are the principal rajlii of curvature and A
is the unit vector parallel to the normal at the extremity of z.
76 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
We now proceed to consider another special system ot
coordinates.
' The 'elements dz and z on the surface which are drawn
through the extremity of the vector z are perpendicular if
dz 8z = 0;
that is, if
cl </#! Sx l -f ^ c 2 (dx l 8x^ + dx^Sx^ + z\ dx 2 Sx 2 = 0,
or a ll dx l 8x l + a l2 (dx i 8x t2 + tlxi8x 1 )+a, i <>dx 2 $x t , = 0.
(56.1)
The elements dz and Sz at the extremity of z are said to be
conjugate, if the tangent planes at the extremity of z and at
the extremity of z + dz both contain the element 8z\ that
is, if Sz is perpendicular to the normals at the extremities of
z and of z + dz. We therefore have for conjugate elements
8zd\ = 0,
that is,
z l \ l dx l Sx 1 + z l \ 2 (d i f\8x 2 -{-(Jx 2 8.r l ) + ^ 2 \^dx 2 8a\,, 0,
or /2 ll rfo' 1 r 1 + n ]t> (djf' l 8x^dx^S^ l ) + /2 2 .//.f 2 <Stf, 2 = 0.
(5G.2)
Thus we see that the lines of curvature at any point of
a surface are both orthogonal and conjugate, and conversely
we see that lines which at any point are both orthogonal and
conjugate are lines of curvature.
An element which is conjugate to itself satisfies the equation
fi ll dxl+2fi n dx l dx 2 + ftM(fa% 0.
The self-conjugate elements at a point form the asymptotic
lines n n dx\ + 2 fl^lx^a^ fl^(lx\ = ; (56 . 3)
and we see that the radius of curvature of a normal section
in the direction of an asymptotic line is infinite.
fix ft f
We call ~ and -.- a the 'direction cosines' of an element
ds ds
on the surface. They tell us the direction but they are not
the cosines of the angles the element makes with the para-
metric lines. We often write them in the tensor notation
I 1 , 2 ; but we must remember - is not the square of , nor
REFERENCE TO ASYMPTOTIC LINES 77
is Qf 1 ) 2 a tensor component 12 , but the square of l . We
have identically ^'i-^i k = 1 ;
and, if R is the radius of curvature of any normal section of
the surface, Rfyk?? = 1- ( 5G 4 )
Take now the asymptotic lines as parametric lines. We have
n n = o, n 2 , = o,
and therefore by Codazzi's equations
/2 u /2. 2 -/2 J2 /2 12 -7va =
l
we have log/2 12 = (111 j- {212],
Now we saw ( G) that the determinant a satisfied the
equations :\
--
o y
and therefore c - (log 7v r ^) -f 2 { 2 1 2 } =0,
^ (log /vi) -f2{121} = 0. (56.5)
These arc the equations which tho coefficients a^ must
satisfy if the parametric lines are to be asymptotic.
If we are given any ground form, and if we could transform
it so that the new coefficients would satisfy the above equations,
then we could, since in this case we would know the functions
/2 U , /2 r> , /2 20 and the ground form, find the surfaces to which
the form would be applicable. But the transformation would
itself involve the solution of differential equations of as
great difficulty as Codazzi's equations.
Taking the asymptotic lines as coordinate axes we have
^A! = 0, z 8 A 2 = 0,
78 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
and therefore z l = ^>AA 1 , (56 . 6)
where p is some scalar.
Similarly we have ^ _ q ^ (56 . 7)
where q is a scalar.
As 3X = A
we have 2)S\X 1 X 2 = qS\\ 2 \ l =
and therefore ^ ~ ~ 7- (56.8)
Since A! A., = Kz^z^
we have - 7^ 2 FAA T AA 2 = A^s ,
that is, -7i^ 2 (A 2 /S t AX 1 X-A&'XA l X 2 ) = C A 2^
or A> 2 XAS r A'^ = X, (56.9)
since A X A 2 is parallel to A.
We therefore have p ( K )~t, (56.10)
and s 1 =(-A r )"*XX l> 2 - -(-A r )-*XX a . (56.11)
These are the exceedingly important equations which we
have when we choose the asymptotic lines to be the para-
metric lines.
67. Equations determining a surface. If we now take
Z=(-K)~*\, (57.1)
so that Z is a vector, parallel to the normal at the extremity
of z y and of length ( A r )"J, we can write the equations which
determine the surface in the simple form
From these equations we have
^12 = 0,
and therefore Z 12 = pZ, (57.2)
where p is some scalar [not the p of (56 . 10)].
In order to find the asymptotic lines of a given surface we
have to solve the ordinary differential equation
v* = 0,
EQUATIONS DETERMINING A SURFACE 79
and when we have done this we can bring the equation of
the surface to the form stated.
We have Z = cA, '(57.1)'
and we notice that c is an absolute invariant.
Differentiating we see that
and therefore cA^ = CupCy
that is, ca' 12 = C 12 pc. (57.3)
From the formulae
a' 12 = fl^f-^, + ITT/ )>
we see that
and therefore p = - /2 12 ^ -f -^ - (57 . 4)
The equation of the surface referred to the asymptotic lines
is therefore z l = ZZ 19 2 = -ZZ^ (57 . 5)
where Z n = -/2 W + ^. (57.6)
68. The equation for the normal vector in tensor form.
We can express the equation which the vector Z must satisfy
in tensor form so as to be independent of any particular
coordinate system.
The null lines on the surface applicable on the ground form
are the lines which satisfy the equation
a ik dx { dx k = 0.
On a real surface they are of course imaginary and are
characterized by the property that the distance, measured
along the curve, between any two points on a null curve
is zero.
Let us now consider the ground form
fl ih d'^dx^ ' (58.1)
80 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
remembering that any quadratic differential expression is the
ground form of some set of surfaces. The surface, to which
this form applies, will have as its null lines the correspondents
of the asymptotic lines on the surface we are considering.
Let Beltrami's differential operator with reference to the
ground form fl^dx^lx^
be denoted by <yA 2 . (58 . 2)
Now we saw (43 . 24) that, with reference to the null lines
as parametric lines, that is with reference to the asymptotic
lines on the surface we are considering,
2.
/2 12 du dv
The equation Z n =
may be written
that is, tt A 2 ^ = (_ 2 ( + - /7 ))^ ; (58.3)
and this is a tensor equation independent of any coordinate
system.
59. Introduction of a new vector We may write
this tensor equation briefly in the form
^Z = pZ. (59.1)
Let be any scalar quantity which satisfies the equation
A a = jp0.
We then have \ K = Z A 2 6.
Now we saw (17.4) in the chapter on tensors that
where ^u f
and therefore u t v l = v t u*.
INTRODUCTION OF A NEW VECTOK 81
We may then write the equation
6A.,Z-Z\0 ^
in the form 6 * VttZ<< = Z --
Z6 ( ) Q (59.2)
[where /2 denotes /2 n /2 22 ~/2i 2 ].
If the asymptotic lines are real /2 will bo negative : we
therefore write this equation
, .- __. c , -!#.{__ #02j_ Ot (59.3)
We can then by quadrature find a vector such that
that is,
V-/2 = 7(/2 1 ^ 1 -/2 ]1 ^)-tf(/2 1 ^ 1 -/2 ll ^),
^-^,,^,)-^^^,-^^). (59.4)
It should ho noticed that to find required a solution of
the equation ^ g _ ^ e ( 59 B 5 )
60. Orthogonally corresponding surfaces. We have
V(A,,Z)Z= 0,
and therefore
or, since
^72/^ = 0. (GO .
We can therefore by quadrature find a vector z such that
z l = V
that is, ^ y
^-/2 I2 ^ 2 ). (60.2)
If the parametric lines are asymptotic these arc just the
equations we began with.
2843 T\f
82 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
We see at once that
3i = 0, 3j/ 2 + 2 = 0, 0^ 2 = 0,
and therefore corresponding elements of the surfaces traced
out by z and by are connected by the equation
dsdf=0 t (60.3)
that is, corresponding elements are perpendicular to one
another. The surfaces are then said to correspond orthogonally
to one another.
61. Recapitulation. We may now restate the results we
have arrived at.
Consider the ground form
and let A 2 have reference to this form. Let Z be a vector
which satisfies the equation
Z) = 0.
Then z l = V, 2
define a surface traced out by a vector z.
On this surface the unit vector parallel to the normal at
the extremity of z is given by
#=cA f
where c ( K)~* 9
and K is the measure of curvature of the surface z.
We have ^Z = Z
. A.,c
where 2 , = __
The asymptotic lines on z are
n ik dxt
The surfaces given by
where 6 is any scalar satisfying the equation
. **6=p0 9
correspond orthogonally to the surface z.
RELATIONSHIP OJ? SURFACES Z AND 83
62. Relationship of surfaces z and When the para-
metric lines are asymptotic on 0, that is, when
n n = 0, /2 22 = 0,
and therefore f M = |* f 2 + -^ . (62 . 1)
The parametric lines on are now conjugate lines : for if
we have an equation of the form
where p and q are any scalars,
Xj l2 = 0.
If p 2 = ? l
the conjugate lines have equal invariants in Laplace's sense.
The parametric lines on are therefore said to be conjugate
lines with equal invariants. To the asymptotic lines on z
there correspond therefore conjugate lines with equal in-
variants on g.
If on any surface we are given the conjugate lines with
equal invariants, we can find by mere quadrature a surface
which will correspond orthogonally to . For if
~ ~
where 6<f> = \
and therefore ~- (<f> 2 ^) + ~ (<^) = o, (62 . 2)
<j 16 o V
where <j> 2 means the square of (f> and is not a tensor notation.
We can therefore find by quadrature a vector Z such that
that is, f l =
The surface given by
84 TWO-WAY SPACE AS A LOCUS IN EUCLIDEAN SPACE
will correspond orthogonally to and will have the asymptotic
lines as parametric lines.
We have now seen the relationship to one another of the
surfaces z and and the method by which, given either, we
are to obtain the other.
63. Association of two other surfaces with a c-surface.
Let a vector m be defined by the equation
We have, taking as parametric lines the conjugate lines
with equal invariants on
and therefore
O^Z-m) = Otfi + mJ, 6. 2 (Z+m) = 0(# 2 -wJ.
(63.1)
From these equations we see that
Z 12 Z o, m J2 m = o,
and, as
V(Z-m)(Z l + m 1 ) = 0, V(Z+m)(Z.,-m^ = 0,
ZZ l luiihi -f ZiUi 4- ^^^ = 0,
J&ziMi z Zm 2 Zs>ti = 0. (G3 , 2)
We can take s, = ^, , z. 2 = ZX^
and we have 2/i ~^i T~"
It follows that y only differs by a constant vector from
s + Zni.
We have thus obtained the surface y, where
2/ = s-{-#m, (63.3)
directly from z and and the asymptotic lines on this surface
correspond to the asymptotic lines on z.
FOUR RELATED SURFACES 85
64. \Vc obtain yet another surface directly from the
definition ?; = #0, ( 64 -J)
where 6(/> = 1 ;
and we see that
so that 77 L m(f> l f in l (f>. (04 . 2)
Similarly we see that
rj^ m 2 (j)~ m0 2 . (G4 .3)
The surface 77 will therefore correspond orthogonally to the
surface y ; and to the asymptotic lines on y will correspond
on 77 conjugate lines with equal invariants.
We have thus four mutually related surfaces,
s> 2/> ^
which are intimately connected with two problems in the
Theory of Surfaces, viz. the theory of the deformation of a
surface, and a particular class of congruences of straight lines.
The relations between the four surfaces will be more com-
pletely stated when eight other surfaces are introduced, as
they will be when we consider the Deformation Theory.
CHAPTER V
DEFORMATION OF A SURFACE, AND
CONGRUENCES
65. Continuous deformation of a surface. We have seen
that the problem of determining the surfaces in Euclidean
space, to which a given ground form
appertains, depends on the solution of the equations
and we have pointed out the difficulty of solving these
differential equations.
There is a related problem the solution of which is simpler.
This problem is the determination of a surface differing
infinitesimally from a given surface and applicable upon the
given surface. Let z be the vector of the given surface, and
z + t the vector which describes the neighbouring surface
which we are seeking, t being a small constant.
We may regard t as a small interval of time and ^ as
a linear velocity vector, descriptive of the rate of increase
of z, as we pass to the neighbouring surface which is applicable
upon the given surface ; or as the growth of the vector z
under the condition of preserving unaltered the element of
length.
If we can obtain we have the vector which defines the
continuous deformation of a surface.
We have at once
i& = 0; X& + * a fi = 0; * 2 &=0, (65.1)
CONTINUOUS DEFORMATION OF A SURFACE 87
that is, the vector describes a surface corresponding ortho-
gonally with the given surface described by z.
An interesting and immediately verifiable theorem on
surfaces which correspond orthogonally is the following :
' If z and { correspond orthogonally, then the surfaces traced
out by z -h and z are applicable on one another ; and
conversely, if z and are the vectors of two surfaces applicable
on one another, z + and z will be the vectors of two
surfaces which correspond orthogonally/
66. A vector of rotation. From the kinematical relation
of the vectors z and wo see that d is the relative velocity
of the extremities of dz in the deformation of the surface z.
In the deformed surface the element which corresponds to
dz will have the same length but will have turned through
an angle. Let the rotation necessary to produce this be
represented by the vector tr.
Now if a vector a, drawn from a point, is made to rotate
with an angular velocity whose magnitude and direction is
represented by a vector r, drawn through the same point, the
linear velocity of tho extremity of a will be given by ra.
It therefore follows that df = rdz,
or 1 = ^1; & = ^V (66. 1)
The vector r is parallel to the normal to tho surface ^, at
the extremity of tho vector g. We therefore have
where a is some scalar ; and therefore
since c^ = ; ^ 2 + ^ = ; 5j = 0,
so that r = ^y; (66.2)
and thus r is uniquely obtained, when z and { are known.
88 DEFORMATION OF A SURFACE, AND CONGRUENCES
67. Geometrical relationship of surfaces traced out by
certain vectors. In exactly the same way we see that
where P=T L ^' ( G7 - 2 )
&Vp2
By differentiation of the equations
we sec that r^ l ~ r,^; (G7 . 3)
and therefore the vectors r t , r 29 y lt z 2 are all parallel to the
same plane. It follows that the normals to the surfaces
traced out by z and r are parallel at corresponding points.
Similarly wo see that the normals to the surfaces traced
out by the vectors p and are parallel at corresponding points.
But the vector r is parallel to the normal at the correspond-
ing point of : it is therefore parallel to the normal at the
corresponding point of p.
From the equations r* ^
we see that rp = 1. (67 . 4)
It follows that the r and p surfaces are polar reciprocals
with respect to a sphere whose centre is at the origin and
radius the square root of minus unity.
68. The angular velocity r is applied at the extremity of
the vector z. Now an angular velocity r, at the extremity
of the vector 0, and an angular velocity r at the origin, are
equivalent to a linear velocity zr.
It follows that a linear velocity and an angular velocity r,
at the extremity of 0, are equivalent in effect to a linear
velocity + zr, and an angular velocity r at the origin. We
are thus led to consider two other vectors,
+zr and
TWELVE ASSOCIATED SURFACES 89
69. A group of operators, and a system of twelve
associated surfaces traced out by vectors. The fundamental
relations between the vectors 0, r, p, arc expressed by the
equations d{=rds; dz = P 7t{. (69.1)
These relations are unaltered by the transformation scheme
in 0, r, p,
*' = ^ + P; f = r; r' = ^ ; p' = s, (09 . 2)
which we shall denote by the operator A.
They are also unaltered by the transformation scheme
' = {; f=; r' = p; p' = r, (69.3)
which we shall denote by the operator S.
We see that the operators A z , A*, A 4 , A'' tiro respectively
the transformation schemes
. /' _
_ P. .
We see that A 6 = 1 ; B* = I, (69 . 4)
and A*B = BA; A*B = BA 2 ; A*B = BA* A 2 B = BA* ;
AB = BA*,
and so the operators A and B form a group of order twelve.
The operators A form a sub-group of order six ; -the opera-
tors /{ form a sub-group of order two.
If we take p = A 3 Q = BA ; R = A 2
we have P 2 = 1 ; Q 2 = 1 ; R 3 = 1, (69.5)
PC = QP; PR = RP;QR = R*Q -, QW = RQ,
and the operators P, Q, R will generate the same group. Of
this group the operators P form one sub-group, the operators
90 DEFORMATION OF A SURFACE, AND CONGRUENCES
Q another sub-group, and the operators P and Q together
a sub-group of order four. The operators R form a sub-group
of order three.
We thus obtain directly from the four vectors 0, r, p
a system of twelve vectors which trace out twelve surfaces
connected in various ways at corresponding points.
70. We may arrange the twelve surfaces in tabular
form thus
The first column will denote a vector of a surface; the
second the vector of the surface which corresponds ortho-
gonally to the surface in the first column and in the same
row; the third column will denote the vector which gives
the angular velocity corresponding to the surface in the same
row but in the first column; the fourth will denote the
angular velocity which corresponds to the surface in the same
row but in the second column.
The vectors in the third column are parallel to the normals
to the surfaces in the second column and in the same row ;
the vectors in the fourth column are parallel to the normals
to the surfaces in the first column and in the 8ame row.
Finally the surfaces in. the same rows and in the third and
fourth columns respectively are reciprocal to one another.
THE TWELVE SURFACES FOKM THEEE CLASSES OF FOUE 91
71. The twelve surfaces form three classes of four. Let
us now recall what wo proved about the four surfaces which
we denoted in 62-4 by 2, , y, 77, and the equations of con-
nexion when z is referred to its asymptotic lines.
We had
mO, Z = 77$, y = z -f Zm.
Wo see that Z is parallel to the normal at the extremity of
z, and p is parallel to the same normal. Therefore
Z = pp, (71.1)
where p is some scalar.
Now z= = O'Z'
but Si = ^i,
and therefore Z dp,
that is, 77 = p. (71 . 2)
Now y = z -f Zm z + rjg,
and therefore y z -f^p. (71 . 3)
The four surfaces are therefore in the present notation
(merely changing the sign of the vector 77)
that is, s, Hz, BAz, A 5 z,
or c, PQRz, Qz, PJlz.
Now the asymptotic lines correspond on two surfaces which
are polar reciprocal to one another, since conjugate lines
reciprocate into conjugate linos; and we know that the
asymptotic lines correspond on
z and z -h p.
The asymptotic lines therefore correspond on
that is, on z, Pz 9 Qz, PQz.
92 DEFORMATION OF A SURFACE, AND CONGRUENCES
The surfaces which correspond to these orthogonally are
respectively ^ ^ + -
that is, PQRz, QRz, PRz, Rz.
On these surfaces there correspond to the asymptotic lines
conjugate lines with equal invariants. We Avill say con-
jugate lines with equal point invariants.
The surfaces which are respectively reciprocal to these
four are ,
r t ^ f+rz
rf { + '' * V'
that is, R 2 z, PR*z, QR*z, PQR 2 z.
We say that on these surfaces there correspond, to the
conjugate lines with equal invariants on their reciprocals,
conjugate lines with equal tangential invariants.
The twelve surfaces thus fall into three classes: viz. those
on which the asymptotic lines correspond ; those on which
conjugate lines with equal invariants correspond ; those on
which conjugate lines with equal tangential invariants corre-
spond. The surfaces of any class are permuted amongst
themselves by the operations of the sub-group
i; P-, Q; PQ.
72. A case in which one surface is minimal. If the
vector z is of constant length we can prove that the surface
+ w (72.1)
is a minimal surface.
We saw that the normals at corresponding points of z and
of r were parallel. If then z is of constant length, the vector
is parallel to its own normal and therefore equal to AA,
where k is a constant, and A is the unit vector parallel to
the normal at the extremity of r. But
and therefore
A CASE IN WHICH ONE SURFACE IS MINIMAL 93
We saw (52 . 8) that the condition that z might be a
minimal surface was # 01 AA 2 = Uz 2 \\ l9
and clearly this condition will remain the same if we replace
X by any vector parallel to it.
Let y = g+zr.
We see by tho table that z is parallel to the normal at the
extremity of y. The condition that y may be a minimal
surface is then Sy^zz^ = Sy^zz^ (72 . 2)
But by the fundamental formula of connexion and by the
table we see that ^ ^
2/i = - r i 2/2 = 7 V
Tho surface y will therefore be a minimal surface if
Lliat IS, it ZZ^ZT-i Z T-tZt) ZZ-tZ^^ Z ^Vj^i
Now z being of constant length this condition becomes
At /" r> .
M^2 'v^i
and this we have seen is true.
This theorem will bo used in proving an interesting theorem
of Ribaucour's in connexion with a particular class of con-
gruences.
We now proceed to consider the theory of congruences of
straight lines in connexion with which the twelve surfaces
will be of interest.
73. Congruences of straight lines. If we wish to con-
sider not merely the geometry on one particular surface but
the relation of points on that surface to corresponding points
on another surface, we are led naturally to consider the
congruence of straight lines which join the corresponding
points.
Let z be a vector depending on two parameters u and v, and
ft a unit vector depending on the same two parameters, and
drawn through the extremity of z. Let iv be a length taken
along the vector p, ; the congruence will, then be defined by
(73.1)
94 DEFORMATION OF A SURFACE, AND CONGRUENCES
We regard z and p as functions of the parameters u and v,
and therefore the current vector z 1 will bo a function of the
three parameters u, v, and w.
The unit vector p will trace out a sphere which we call
the spherical image of the congruence.
Let
so that dcr* = a u dfu a + 2a l2 dudv + a 22 dv* (73 . 2)
is the ground form of the spherical image.
Let Wk = <*ik>
ll> ^n a 2U ""^12
and notice that in general
>ik = *>li-
If we take two neighbouring rays of the congruence we have
dz' dz + wd/j. -f p. du\
If X is a unit vector perpendicular to p, and cZ//,
da\ = pit //,
and therefore
Jul<r\
It follows that
<o n dv) (a l2
But, if 5 is the shortest distance between the two neighbour-
ing lines, 5 -AcZc,
and therefore
(73.3)
a T T du + a . 9 r /?;, a, cZu 4 a 99 dv
11 ' 1 / 7 14 Z^
74. Focal planes* and focal points of a ray. The value
of w which corresponds to the shortest distance between two
FOCAL PLANES AND FOCAL I'OINTS OF A RAY 95
neighbouring rays ivS given by the fact that dz' is perpendicular
to \L and fi -f d/jL ; and therefore
dz'dp. = 0.
We thus have dzdp + wdfi 2 = 0,
o) u du 2 -f (o) 12 -f eO dudv 4- co^dv 2 "
~ll - ~ - - --
,, ,
so that
( 74 .1)
'
The critical values of w, say w' and tc", as we vary the
ratio du : dv, are therefore given by
co n wet
-f ox,,) wu
- 0, (74 . 2)
and the corresponding values of the ratio du: dv are given by
= 0.
(74.3)
There are, by (73. 3), two values of the ratio du:dv which
make 5=0. Through each ray of the congruence there thus
pass two developable surfaces defined by
= 0.
(74.4)
The planes which ])ass through this ray and touch the
developablcs are called the focal pldiies of the ray. The
points where the ray is intersected by these neighbouring
rays are called the focal points of the ray.
The developables are defined by
= p (a^d
^du +
where p is some multiplier; and we see that this multiplier
is w. The focal distances, /' and /", are therefore the values
of w which satisfy the equation
(74.5)
06 DEFORMATION OF A SURFACE, AND CONGRUENCES
75. Limiting points. The Hamiltonian equation. Prin-
cipal planes. If we have any two real quadratic forms
we can, by a real transformation, bring them to such a form
that in the new variables
It is therefore possible by a real transformation to make
12 = 0. (75.1)
The points on the ray given by w', w" are called the
limiting points of the ray. These points are therefore real.
If we suppose the transformation applied which makes
we have o> n = w'<t UJ o> 22 = iu"a 22 ;
and the value of w which corresponds to the shortest distance
between two neighbouring rays is given by
?(/(?!, du' + iv"a 29i dv 2
We may take
COS 2 e - j^Li^ 2 S m 2 e = -A 2 -^ 2 ,,,
a n du* + <t 2< ,<lv* (( n du* + ^^dv L
and we have the Hamiltonian equation
w = ti;' cos 2 + ^ r/ sin 2 (9, (75 . 2)
showing that the shortest distance between any two neigh-
bouring rays lies between the two limiting points.
The values of the ratio duidv which correspond to the
limiting points are given by
(w'-w")dudv= 0.
Leaving aside the special congruence when the limiting
points may coincide, we see that corresponding to the limiting
point w\ du is zero, and the shortest distance is parallel to
fjL/ji 2 . Similarly the shortest distance corresponding to to" is
PRINCIPAL PLANES 7
parallel to /i/^; and these shortest distances are perpendicular
to one another since
^>/*iW*a = W*Wi-PlW* = - ( 75 3 )
The planes through the ray /z which are perpendicular to
these shortest distances are called the principal planes of the
ray : and they are perpendicular to one another.
70. Principal surfaces, and the central surface. Return-
ing now to general coordinates we see that
and therefore, in the important class of congruences for which
co 12 c 21 , the limiting points and the focal points coincide.
We see also that the focal planes will then coincide with the
principal planes.
When we take any equation connecting the parameters u
and v of the congruence we obtain a ruled surface of the
congruence. The directrices of the ruled surface will be
curves lying on the surface z. If u and v are functions of
a variable p, then p and w will bo the coordinates of the
ruled surface. The lines of striction on the ruled surface
will bo given by
~ cfo8 (76.1)
where u and v are connected by the equation which defines
the ruled surface.
The ruled surfaces given by
-f o> 21 ) db\ ^ (co V2 -f o> 21 ) da -f o>^^ v>
u + a n dv, a^dvu + a^dv
(76.2)
are called the principal surfaces of the congruence.
The locus of the points on rays midway between the foci,
and therefore midway between the limiting points, is called
the central surface of the congruence.
98 DEFORMATION OF A SURFACE, AND CONGRUENCES
77. The focal surface. Any ray of the congruence will
be intersected by a neighbouring ray if
dz + wdp + pdw = 0.
The developables which pass through the ray are therefore
given by Sdzdpfi = ;
that is, by S (z^lu, + z^lu) (fadu + fadv) fi 0.
The focal points are given by
(z l + W/JL^ du 4- (^ 2 + w/O d v + pd = 5
that is, by S (z^ + W/JL^ (z% 4- w// 2 ) /* = 0.
The focal surface of the congruence is defined as the locus
of the focal points on the rays of the congruence. If we so
choose the parameters that the equation defining the develop-
ables is dudv = o,
then Kz l jji l p. o, Sz^fajj. = ;
so that o 1 = a ^ + 6//, z> c^ + dp,
where a, 6, c-, d arc scalars.
Substituting in the equation
S (z l -f wfij) fa + WfjiJ n = 0,
which defines the focal points, we see that the focal surface
has two sheets given by
Z' = Z (l/JL, Z* = 2C/JL.
78. Bays touch both sheets of the focal surface, The
congruence of rays of light. For the first sheet
z\ = (I - x ) p, z\ = (c - a) p z + ((Z - ei 2 ) //,
so that the normal to the first sheet is parallel to /JL/JL^ ; and
the ray touches the first sheet along the u curve on it that
is, the curve along which only u varies ; and the v curve is
conjugate to the u curve.
Similarly wo see that the ray touches the second sheet
along the v curve on it, and the u curve on it is conjugate
to this.
Thus any ray of the congruence touches both sheets of the
THE CONGRUENCE OF RAYS OF LIGHT
99
focal surface ; and the tangent planes to the focal surface at
the two points of contact are the tangent planes to the
developables through the ray.
The edges of regression of the developables are the u curves
on the first sheet, and the v curves on the second sheet.
If the congruence is formed by rays of light, the focal
points on the ray are the foci as defined in the theory of thin
pencils. F l and F 2 arc the foci on what is called the principal
ray of the thin pencil. The tangent plane at f^ to the
second sheet, which is the tangent plane at F l to the develop-
able, is called the first focal plane : so the tangent piano at F l
to the first sheet, which is the tangent plane at jF 2 to the
other developable, is called the second focal plane.
The devolopables through any ray are somewhat like the
above figure.
The focal lines as defined in some text-books on Geometrical
100 DEFORMATION OF A SURFACE, AND CONGRUENCES
Optics have no meaning at all; but it has been pointed out
that the lines conjugate to the principal ray on each sheet
have a physical meaning which might entitle them to the
name of focal lines.*
79. Refraction of a congruence. Malus's theorem.
A congruence is given in terms of the coefficients a^. of its
spherical image and of the coefficients
o> n , o> 12 , o> 21 , co 22 ,
We can see how the congruence, when we regard it as
formed by rays of light, is altered by re-
fraction at any surface z, whose normal is
parallel to the unit vector A.
Let p be the unit vector into which /z is
refracted : that is, let // trace out the new
spherical image.
We have /*' = f*/z + 6A, where a and b
arc scalars. In the ordinary notation of
optics, if cf) is the angle of incidence, 0'
the angle of refraction, and k the index of
refraction,
k sin 0' sin 0.
Now X/JL' (iXfj., P-'P' bXji;
and therefore
a sin $ = sin0', 6sin0 = sin (0 0'). (79 . 1)
We thus see that a is a constant independent of the angle 0,
but b depends on 0. We have
A/z-f cos0 0.
Since /// = a/j, i + IXj + b { A,
we have &>;//
where /2^ refers with its usual meaning to the surface of
refraction.
* [Probably the allusion is to a note 'On focal linos of congruences of
rays' : Elliott, Messenger of Mathematics } xxxix, p. 1.]
REFRACTION Otf A CONGRUENCE. MALUS's THEOREM 101
We see that cos 0' = a cos + 6,
and therefore if we multiply
by /JL', that is, by
and take the scalar product, we get, since fi'ft'j is zero,
(tbiXpi + kjji) = /^cos0'. % (79.2)
We notice that if o^., co a
then o>' r , fc^r
We shall see (83.2) that the condition
o> la = o> 21 (79 . 3)
means that the rays of the congruence arc normal to a system
of surfaces and we now see that this property is unaltered by
refraction. This is Malus's theorem.
We have now given the equations which would determine
any refracted congruence, when we are given the refracting
surface. Unfortunately the equations are complicated.
> 8O. The Ribaucourian congruence. We shall now con-
sider some special classes of congruences.
Consider the congruence formed by rays drawn from every
point of a surface, parallel to the normal at the corresponding
point of a surface which corresponds orthogonally to the
given surface. This is the llibaucourian congruence, so called
as Kibaucour was the first to consider it.
We take to be the surface from which the rays are drawn
parallel to the normals to the surface z.
Taking the asymptotic lines on z as the, parametric lines
we had f A ZQZ f QZ QZ
Sl l/i*> I'"]) &2 u "2 l/ 2 /V >
and Z c\,
where c = ( K)~*,
K being the measure of curvature on z.
To bring this into accordance with our notation for con-
gruences we write p. for A, and we hav
102 DEFORMATION OF A SURFACE, AND CONGRUENCES
Since S^fa/i and $^ 2 // 2 // = 0,
the equation which defines the developables is
dudv = ;
and the local points are given by
w = cO, w = cO.
The surface is then the central surface of the congruence,
and the developables intersect it in conjugate lines with
equal invariants. These lines correspond to the asymptotic
lines on z, the surface which corresponds orthogonally to the
central surface.
81. The Isotropic congruence. Ribaucour's theorem.
We have a particular, and most interesting, case of this con-
gruence, when the surface which corresponds orthogonally
with { is a sphere with the origin as centre.
In this case c is a constant and { corresponds orthogonally
with p, itself.
The congruence is z' = + w/t
and is called the isotropic congruence.
For the isotropic congruence,
and therefore the limiting points of any ray coincide and are
on the central surface. Any plane through a ray is a prin-
cipal plane and any surface may be regarded as a principal
surface. The lines of striction of all the ruled surfaces of the
congruence lie on the central surface.
In the chapter on the ruled surface [see 108] we prove
that any two ruled surfaces of the congruence intersect at
the same angle all along their common generator.
The dev.elopables and the focal points we see are imaginary.
We have proved that y = s+ ^
is a minimal surface and that p. is the unit vector parallel to
the normal at the extremity of y. The perpendicular p
on the tangent plane to this surface is given by
p + yp = 0,
* - '
that is, by p + IL ~ 0.
ISOTROPIC CONGRUENCE. RIBAUCOUR's THEOREM 103
The tangent plane is therefore the plane drawn through the
extremity of perpendicular to the ray of the congruence.
We thus have Ribaucour's theorem that c The envelope of the
plane, drawn through the extremity of the vector which
traces out the central surface, perpendicular to the corre-
sponding ray of an isotropic congruence, is a minimal surface '.
The surface corresponding orthogonally to the sphere is
therefore the pedal of a minimal surface.
If two surfaces are applicable on one another, and if the
distance between corresponding points is constant, we see
that the line joining these points traces out an isotropic con-
gruence. For if fj, is the unit vector parallel to the join of
the points, and z is the vector to the middle point of the
join, and 2c is tlie length of the joining line,
from which equations we at once deduce the result stated.
82. W congruences. Let us now consider again the two
surfaces which we denoted by z and z + gp y and consider the
congruence formed by the line joining corresponding points
on these surfaces. Looking at the tabular arrangement of
the twelve surfaces we see that p is parallel to the normal
to z at the corresponding point, and that is parallel to the
normal to z + p at the corresponding point. The line joining
corresponding points on the two surfaces z and z + p, being
perpendicular to both p and is perpendicular to the normals
to z and to z + p, and therefore touches each of these surfaces.
Now if a ray of a congruence touches a surface, that surface
must bo a focal surface of the congruence. For, taking z to
be the vector to the surface, and p the unit vector parallel
to the ray, 8^*2 = 0;
and therefore, the focal points being given by
S (z l + w^} (z 2 + Wfi 2 ) 14 = 0,
we see that one of the focal surfaces is given by w = 0.
101 DEFORMATION OF A SURFACE, ARD CONGRUENCES
It follows that z and z + p are the focal surfaces of the
cpngruence we are considering.
Now on these surfaces the asymptotic lines correspond.
Conversely it may be shown, that if the asymptotic lines
correspond on the two sheets of the focal surface the focal
surfaces are z and z + gp.
Congruences of this type may be called W congruences.
83. Congruence of normals to a surface. We now come
to the case of congruences where the rays are normal to
a surface. The theory of such congruences is of special
interest in geometrical optics as well as in geometry.
Instead of /z we shall write X, where A is the unit vector
normal to the surface from which the rays emanate.
AVe now have ^ ^ = ^\ (83.1)
as a necessary condition that the congruence may be a
normal one.
This necessary condition is also sufficient : for if
n
then T-/*^ N ~/* i>
^H,^* <*V^-
and we can therefore determine a function w such that
n\ = /^ ; u' 2 = /*c. 2 .
Let z' = z + iup.,
then z\yi = ZIIL + U\I& = 0,
so that the rays are normal to the surface z'.
The normal congruence is therefore defined by
o> 21 = a> 12> (83.2)
and the limiting points coincide with the focal points, and
the focal planes with the principal planes. The focal planes
are therefore perpendicular to one another.
Conversely if the focal planes are perpendicular to one
CONGRUENCE OP NORMALS TO A SURFACE 105
another the congruence is a normal one : for we see that the
condition that the focal planes may bo perpendicular is
and therefore, since <-i n <(^ 2 a\ a is not zero,
o> 12 = o> 2l .
84. Reference to lines of curvature. We now take the
parametric lines on the surface z to be the lines of curvature,
and we have z l = -R f \ 1 , ^=-R"X t , t
where II' and R" are the principal radii of curvature.
We have
co n = -R'\l a> la = a> al = 0, a> 2a = -.K"A 3,
that is, o) u R'a lv a>^ = R"a^, co^ co zl u ri 0.
The focal points are given by
f=R', f' = R",
and the two focal surfaces are now given by
The equation of the developables is
(R'-R")dudv = 0.
As we need not consider the case where R' R" any
further than we have already done we see that the equations
of the developables are
du = 0, tZy = 0. (84.1)
For the focal surfaces we have
</o' = -(R"-R')\dn + \dR'. (84 . 2)
Calling this the first sheet of the focal surface, its ground
form is (dR')* + (R"-R')*Mdv\ (84 . 3)
and therefore the u curve is a geodesic on the ih-st sheet.
Similarly we see that the v curve is a geodesic on the second
sheet.
86. Tangents to a system of geodesies. Conversely if
we take any surface, and draw any singly infinite system
of geodesies on it, the tangents to these geodesies will generate
a normal congruence.
106 DEFORMATION OF A SURFACE, AND CONGRUENCES
For take a surface with tho ground form
and consider the congruence formed by the tangents to tho
curves v = constant, that is, by the tangents to this family
of geodesies. We have /z = s l and as
~ 2 _ i ^ ^ n
"i l j ^2 u >
we must have c t : 12 = 0, Sn^ + 'i^iu = 0,
so that C 2u ^ 0.
Now c; 2 /x 1 = o. 2
and c^ = c^ 1
so that ^! = ^ 2 ,
and the congruence is a normal one.
86. Connexion of W congruences which are normal with
W surfaces. Now let us consider the asymptotic lines on
the two sheets of the focal surface.
The vector to the first sheet is
and we have
o' 1 =-R' 1 X, c' 2 = (II" R')
and therefore (R r - R") X 12 = R'\ A 2 - R'
The equation of the asymptotic lines is
if V is the unit vector parallel to the normal at the extremity
of z'.
Now A x is parallel to V, and therefore the equation of the
asymptotic lines is dz'd\ = ;
that is, S((R"-R') \ 2 dv-\dR') (\ u du + X l2 dv) = 0.
We have, since AjA 2 is zero,
A n A 2 = A T X 12 = ^ A 'f -r (R f R")>
XX n = X^, XX ]2 = 0,
W NORMAL CONGRUENCES AND W SURFACES 107
and therefore the equation of the asymptotic lines on the
first sheet is X*R' l d<u,*-\lR" l <W = 0. (8G . 1).
Similarly we see that the asymptotic lines on the second
sheet of the focal surface are given by
\lR' 2 du 2 -\lR" 2 <lv 2 = 0. (86.2)
The necessary and sufficient condition that the asj mptotic
lines on the two sheets may correspond is therefore that R f
and R" may be functionally connected.
We thus have the theorem that in a TF congruence, if it is
also a normal one, the surfaces which intersect the rays
orthogonally have their radii of curvature functionally con-
nected : that is, they are W surfaces.
87. Surfaces applicable to surfaces of revolution, and W
normal congruences. We saw ( 84) that the ground form of
the first sheet of the focal surface of a normal congruence was
(dR'Y + (R"-R')*an<lv\ (87 . 1)
and similarly we see that the ground form for the second
sheet is (dR")* + (R"-RJa n du 2 . (87 . 2)
If the congruence is also a W congruence we know that
the ground forms of the first and second sheet are then
respectively (<fcfl')2 + ^ (R'jfdtf, (87.3)
(dR")* + (<t>'(R')Y*dtf. (87.4)
The two sheets are therefore applicable on surfaces of
revolution, the u curves on the first sheet corresponding to
the meridians, and the v curves on the second sheet.
Conversely, if we have any surface applicable on a surface
of revolution, the curves which correspond to the meridians
will be geodesies, and the tangents to these curves will
therefore trace out a normal congruence which will be a W
congruence ; and the surfaces which cufc the rays orthogonally
will bo W surfaces.
108 DEFOKMATION OF A SURFACE, AND CONGRUENCES
If the surface is one of constant curvature we need to solve
an equation of Riccati's form to obtain the curves which
correspond to the meridians, but in other cases we can find
the curves by quadrature.
An interesting property of any given W surface, which is
not of constant, curvature, is that we can find the lines of
curvature on it by quadrature.
For we can find the two sheets of the focal surface, and on
these sheets we can find by quadrature the curves which
correspond to the meridians. These curves will have as their
correspondents on the given W surface the lines of curvature.
This theorem was discovered by Lie.
88. Surfaces of constant negative curvature. Returning
to the ground forms of the two sheets of the focal surface
we see by aid of the formula
whim the ground form is (hr + ffidr 2 , that, K' denoting the
measure of curvatuie on the first shoet,
Similarly we find for the measure of curvature K" of the
second sheet K" + <^> </') J 4 -f- [(0 (R'))'' <//' (R')] ^ (88 . 2)
since K' = /('), R'-f(R' } = |^
If the two sheets are applicable on one another at correspond-
ing points we must have K' = K" and therefore we must have
0" (IV) (j> (K) - (f (R'))*. (88 . 3)
Taking the upper sign wo see that
where a and b are constants.
SURFACES OF CONSTANT NEGATIVE CURVATURE 109
We now see that R"-R' -a, (88 . 4)
from the equations It' -f (R f ) =
R" =f(R').
The measure of curvature is found to be a"" 2 from the
formula 6" (R'\
A " + ?wr - (S8 - 6)
The two sheets have then the same constant negative
measure of curvature a~ 2 , and the distance between the
corresponding points is equal to the constant a.
We therefore see how, when we are given a surface of
constant negative curvature, we can construct another surface
of the same constant curvature. We find a system of geodesies
on the given surface this involves the solution of an equation
of Riccati's form- and draw the tangents and take a constant
distance a along the tangent: the locus of the point so
obtained will be the surface required.
CHAPTER VI
CURVES IN EUCLIDEAN SPACE AND ON
A SURFACE. MOVING AXES
89. Serret's formulae. Rotation functions. Let A, p, v
be three unit vectors drawn through the origin, respectively
B
parallel to - the tangent, principal normal and binormal of
a curve. We see from the figure that
d\ = /zc/6, dv = /jidrj,
where de and drj are the angles between neighbouring positions
of the tangents and osculating planes respectively in the
sense of the figure.
We thus* have
where the dot denotes differentiation with respect to the arc
of the curve, and p and a- are the radii of curvature and
torsion respectively. We thus have
P -
1
t
a
SERRETS FORMULAE. ROTATION FUNCTIONS 111
and therefore, since X/JL = 0, p.v = 0,
we have /iX = - , jw = --- , ftp = 0.
- ' p - o* < '
It follows that
v^-P- (89.1)
p
These are the formulae of Serrct.
If we were to take unit vectors through the origin mutually
at right angles, the first, X, parallel to the tangent to the
curve, and the second, /z, making an angle (/> with the principal
normal, wo could easily deduce that
X = /jLT vq, /i = i>^ X/', v=- Xq fJLpy
. ; 1 sin cos
where p </> + , q = -- r , r - -
o- p p
More generally, if X, /^, v are three unit vectors mutually at
right angles which are given angular displacements
pds, qds t rds,
we have
\ p.r-~vq, fi = vl> \r, v Xq p.^), (89.2)
as we see from the figure.
The functions p y q, r may be called rotation functions.
If cods denotes the angular displacement which the vectors
regarded as a rigid system receive, where
we can write our equations in the more elegant form
X = coX, (JL = a)//, v = <>v. (89 . 3)
90. Codazzi's equations. It will be useful to consider
a more general displacement.
Let the vectors X, /*, v regarded as a rigid system receive
three angular displacements
co'du, ("dv, co'"dw.
We then have
X 1 = o/X, X 2 = oFx, X 3 = a/^X ;
and therefore co'X^, + o?^X = ^'X^^ 4- a)'\X ;
112 CURVES IN EUCLIDEAN SPACE
or Wo/ 7 *- Fo>"o7x = V(<o'\ -cog X,
that is, Fo/^/'X = F(a) // 1 -a)' 2 ) X. (90 . 1)
Wo have exactly the same equation for p. and therefore wo
have identically ^// = ^ _ ^
Similarly we obtain two other vectorial equations, and we
have
o)'" 2 o>" 3 = o>" o>'", o>' 3 co'"! = a/"**/, a/'j a/2 = a)'a>".
(90.2)
Suppose now that the vectors X, p,, v instead of being drawn
through the origin are drawn at the extremity of the vector z }
which depends on the three parameters u, v, w. If we regard
the extremity of the vector z as the new origin then we may
say that the linear displacements of the origin are
z^lu^ z< 2 dv y z.^dw.
Let -'X +
We therefore have
so that
Similarly we obtain two other sets pf equations :
CODAZZiS EQUATIONS 113
If we ignore the parameter w, we have the six equations :
C*-C\=l>'*l"-P"l' + l"t'-<l't"' (90.3)
These are the equations of Codazzi of which Darboux
makes so much use in his Theory of Surfaces.
91. Expressions for curvature and torsion. Returning
now to the case of a curve, Serret's equations may be written
where
A = coA, fi COJJL, v cov ;
>- A "
00 "~ a- p
If is the vector which describes the curve to which we
arc applying Serret's equations we may write
(91.1)
where i ', t", i" are three fixed orthogonal vectors through
the origin, so that z' , z" , z'" are the Cartesian coordinates of
any point on the curve.
We have z ~ A and therefore
or .
a pz y v <rpz + <rpz + 8.
P
Denoting the components of the vectors A 3 /z, v with respect
to i \ i"> if" in the usual way, wo know that
" '"
A', A", A
and therefore
and
v , v
z', z", z"
z', z", I''
z', s", z'
(91.2)
1
p*
(91.8)
These are the usual formulae in the theory of curves.
2813 Q
114
CURVES IN EUCLIDEAN SPACE
If we take, as is more usual, x, y, z to be the Cartesian
coordinates of any point on the curve and regard them as
functions, not of the arc, but of any variable, we see that
, y, 1 * "2
// <j / /
i/'j -U , *
/y, *
(91 .4)
(91.5)
4 , & *
, y, ^
92. Determination of a curve from Sorret's equations.
We must now show how the equations
.__/* ._ X v ___/*
p ' p cr y <r
determine the curve when we are given the natural equations
of the curve ; that is, when we are given p and <r in terms of
the arc.
Any unit vector may be written
sin cos $ . X + sin 6 sin (f> . p. f cos . v.
Expressing a fixed vector in this way, and noticing that
there can be no relation between the vectors A, p,, v of the form
pX + qp + rv = 0,
where p, q, r are scalars, we find, by aid of Serret's equations,
DETERMINATION OF A CURVE FROM SERRET's EQUATIONS 115
Let ^ = cot I e"/',
u
then we find that \/r = L (\// 2 I) ^- (92.2)
This is an equation of Riccati's form. When we have
solved it, we know & and 0, and thus the position of a fixed
vector with reference to A, /z, v. When we have thus found
three fixed vectors, with reference to A, /z, v, we know
A, /z, v in terms of the arc.
When we have obtained A in .terms of the arc we can
find z by aid of the equation \ t (92.3)
It must now be shown how, when we are given any curve
in space, any other curve, with the same natural equations,
can by a mere movement in space be brought into coincidence
with the given curve.
IF A , /z , */ denote the positions of the vectors A, /z, v when
the arc s is equal to s or, say, to zero, then we see, by
repeated applications of Serret's equations, that
^c'Ao + e'Vo+c"'^, (92.4)
whero the coefficients of A , /z , */ are known series in powers
of 8.
By a mere rotation wo can bring A , /z , z/ into coincidence
with the tangent, principal normal, and binormal at the point
from which we measure the arc on the given curve.
It follows that A, /z, v will be unit vectors coinciding with
the directions of the tangent, principal normal and binormal
at the point s on the given curve.
A mere translation will therefore bring tne cfirve into
coincidence with the given curve when the required rotation
has been carried out, since we have
z - A, z 9 = A,
> *
and thus c'~0 + a
where a is a fixed vector, that is, a Vector not depending
on the arc.
116 CURVES IN EUCLIDEAN SPACE
03. Associated Bertrand curves. The right helicoid.
.Let us now consider the curve defined by
z' = z + kii. (93.1)
where k is some function of the arc, and let us find the
conditions that the two curves defined by z and z* may have
the same principal normal.
We have
y = (x+pk + k-
X\\ ds
and therefore
A' = (\ + pk + k(^-
X A 6#S
p')dj
Since
XV = 0,
we must have k equal to zero [i.e. k a constant].
Again, differentiating with respect to the arc ',
and therefore /cA( TV ) + ( 1 -ITT? >
*\db / \ /ds*
-
p
2
) - T- 2
a- ds 2
( I Q ^/ ^ ^
Eliminating , and y-^ wO obtain
p A; p or
/; &'
and integrating we have " -f = 1, (93 . 2)
where k' is a constant introduced on integration.
A curve satisfying the above equation is called a Bertrand
curve. We see that the property of a Bertrand curve is to be
associated with another Bertrand curve having the same
principal normal, the distance between corresponding points
being the constant k.
If a Bertrand curve has more than one corresponding curve
ASSOCIATED BERTRAND CURVES. THE RIGHT HELICOID 117
it will have an infinite number of such curves and will clearly
be a circular helix, for p and cr will each be constant.
We can immediately deduce that the only ruled minimal
surface is the right helicoid. For consider the curved asymp-
totic line on a ruled surface. We know that the osculating
plane of any asymptotic line on any surface is a tangent
plane to the surface. The generator of the ruled surface
therefore lies in the osculating plane of the other asymptotic
line through any point on it. If the surface is a minimal one
it must therefore be a principal normal, and since an infinite
number of asymptotic lines cut any generator orthogonally
the asymptotic lines must be circular helices. The surface is
therefore a right helicoid.
94. A curve on a surface in relation to that surface.
We now pass on to consider the curves which lie on a given
surface. Since such curves are defined by a relation between
the parameters u and v, and since z y the vector of the given
surface, is a function of these parameters, we are really
given 2 in terms of one parameter along the curve defined by
an equation F (u> v) = 0.
But since we want to consider the curves in relation to
the surface we proceed by a different method.
We have the formulae
where X is a unit vector parallel to the tangent to the curve,
fji a unit vector parallel to the normal to the surface and
making an angle with the principal normal to the curve ;
and we have seen ( 89) that
sn
p
where p and a- are the radii of curvature and torsion of the
curve.
We know that
118 CUEVES IN EUCLIDEAN SPACE
we can therefore easily verify the formulae
and from these formulae we deduce
///^ 1 1 = (e/2 22
It follows that
and that
A/cs = h
(94.2)
(94 3)
But
and
and therefore
cos 6
(JL\ r L
p
fjLZ ltfi\ =
/2 n -i6 2 -f 2f2 12 uv + f! 2
(94.4)
(94 . 5)
Wo have thus expressed the two angular velocity com-
ponents p and r of the curve under consideration in terms of
the derivatives of the parameters u and v with respect to the
arc and the functions e, /, g and fl n > /2 12 , /2 22 .
We must consider the remaining component q.
As the vectors X, //, v are displaced from their positions
at P to their positions at P', a neighbouring point of the
curve under consideration, we may consider that they are
displaced along the geodesic PT and then along the geodesic
TP 9 .
As we pass along Py the displacement qds is zero and as
we pass along TP' the displacement qds is also zero. The
CURVE ON SURFACE IN RELATION TO THAT SURFACE 119
total displacement qds is therefore just the angle P'TM:
that is [39] i
g= (94/6)
Pff
since the geodesic curvature of the curve is defined by the
formula
Pff
= Lt,
P'TM
We should notice that unlike p and r the angular velocity q
depends on the first ground form only and the derivatives
of u and v and not on 4f2 n , /2 r2 , /2 22 .
We have proved earlier (36 . 3) that
We express this formula in a more convenient form for
some purposes without the aid of the differential parameters by
P, {,t 11 Fl-2a llj F i ^ + a M ^}
where F(v,v)
is the equation of the curve, or, since
^16 + 1^=
and F } (u+{ll\} u*+2 {121}^+ [221] v 1 }
+ F tji (i)+ {112}u 2 +2 {212} uv+ {222} v*)
+ F. n u* + 2F.uUV + F. 22 v* = 0,
and a n
in the form
(94 . 8)
= h
Pa
V,
(94 . 9)
120 CUKVES IN EUCLIDEAN SPACE
Wo have thus found expressions for the angular velocities
; 1 sin cos0 .^ A x
,, = # + -, V = -f. 'r=-f> ()
along the curve in terms of the derivatives of u and v and
the functions which define the ground forms. We notice
that p and r depend only on the first derivatives, but q
depends on the second derivatives and is the geodesic curvature
of the curve.
We have seen [ 49] that the curvature of the normal
section of the surface in the direction of the tangent to the
curve is given by
We thus have Meunier's theorem that
* = i- (*. 10)
p It
The expression -}- % (94.11)
is the same for all curves having the same tangent at the
point under consideration. It is therefore the torsion of
the geodesic curve which touches the curve at that point.
95. Formulae for geodesic torsion and curvature. We
can find another formula to express the torsion of the geodesic
by aid of the formula already proved
& + (R 9 + R") *p. + H'H"jP = o.
Since z = X and (i = i/p Ar,
we have 1 - (R f 4- R") r + R' R" (p* + r 2 ) = 0,
that is, p* + (F ~ r ) (JET ~ r ) = * (95 !)
If we take the parametric lines as the lines of curvature, so
that _ cos 2 Q sin 2
T ~~W~+ R" '
this becomes p = cos sin d (-^ ^777 j
or -h - = cos 0sin0(-g> jp)' (95.2)
FORMULAE FOR GEODESIC TORSION AND CURVATURE 121
Since q is the angular velocity about the normal to the
surface, as we pass along the curve we are considering, we
see that
< t
where q f it, + q" v
is the angular velocity about the normal of the rigid system
made up of the normal and the tangents to the two lines of
curvature.
We thus have the formula for the geodesic curvature
l - = - 6 + (/ (L + q"v. (95 . 3)
P<J
We have
and therefore r Zpq (95 . 4)
depends only on the first derivatives of the parameters u and
v, and so is the same for all curves on the surface having the
same tangent at the point under consideration. This theorem
is due to Laguerre.
In connexion with the formulae
1 sin tan
where R is the radius of curvature of the normal section of
the surface in the direction of the tangent to the curve, it is
useful to remember that if a particle describes a curve on any
surface with velocity V, the acceleration normal to the path
72
and tangential to the surface is --
Pff
96. Surfaces whose lines of curvature are plane curves.
So far the curve we have been considering has been any
curve on the surface: suppose now . that it is a line of
curvature.
122 CURYSS IN EUCLIDEAN SPACE
'X,
We 6a*ve p = 0, as we see from the formula
'
and therefore + - ^ 0. (96 . 1 j
0"
If therefore the line of curvature is a plane Qurve its plane
makes a constant angle with the surface all along it ; and
conversely if the osculating plane at each point of a line
of curvature makes the same angle with the surface the
line of curvature is a plane curve.
We now propose to find the form of a surface if all its lines
of curvature are plane curves.
Let a be a vector perpendicular to the plane line of curva-
ture along which only v varies so that a depends on u only.
Similarly let ft be a vector perpendicular to the plane line
of curvature along which only u varies.
In accordance with our general notation in the theory of
surfaces, let A be a unit vector normal to the surface at the
extremity of the vector z.
We have, since the parametric lines are lines of curvature,
and as as. 2 0, /J^ = 0,
we also have aA 2 = 0, /?A 1 = 0.
It follows that oc=pX l -\-qX, /8 = rA 2 -fsA,
where p, q, r, s are scalars.
We thus obtain the two equations
l = 0. (90 . 2)
Now since SX U X 1 X 2 = 0,
as the lines of curvature are conjugate lines,
? 2 = 8 i = 5
and as there can be no relation between Aj and A 2 of the form
g = 0,
SURFACES WITH ONLY PLANE LINES ftF CURVATURE 123
where a and b are scales, we must hav$A
B = i, * = .'.' (9G.-3)
p r r p
8(7
It follows that (log>) 12 = (logr) 12 = --' ,
and therefore we may take
p = F(u)e', r=f(v)e,
q = F(u)J0 l9 8=f(v)e'0 2 . (00. 4)
We also have 12 -f O l 2 = 0,
so that \ 2 +02 A i + 0i X 2 = ' (96*5)
and 0^ + 0^ = 0. (96.6)
Let us now start again with these two equations.
We see that 0i 2 + M 2 =
tells us that e = f (u) + </> (v) ;
and, since \ } A 2 = 0,
the lines of curvature being at right angles, the equation
tells us that (A?) + 2 2 A = 0,
so that A!+6- 2tf jP(w) = 0,
A^+e~ 2 ^0(v) = 0. (96.7)
We can now so choose the parameters that
Xf+- 8 =0, A+- M = 0.
The spherical image of the surface is therefore given by
where A~ l
U being a function of u only and F a function of v only.
124 CURVES IN EUCLIDEAN SPACE
But, from the expression for the measure of curvature of
the surface c fa* = A 2 du 2 + B*dv
An * /I ^7^\ * /I *A\
KAB + --(-_) + -- ( - - ) = 0.
<)u\A <>u/ 3v\B <>v/
We must therefore have
and therefore 1 = UU-U*+ VV- F 2 + UV+ VU. (96.8)
It now easily follows that without loss of generality we
may take fj C0 sec a cosh u,
V = cot a cos v,
e cosh u- cos a cosy
so that c = - : -- (96 . 9)
sin a v /
If p is the perpendicular on the tangent plane to the
surface of which we have found the spherical image we have
2) + Xz = 0.
It follows that
since A^ = 0, Xz 2 = 0, A r c 2 = 0, A 2 x? 1 = ;
and therefore ^) 12 -I- fl^^ 4- fl^ = 0,
that is, Bince d 12 + ^ t 2 - 0,
pe e =U+V, (96.10)
where [7 is a function of u only and V a function of v only.
We know that
, fi
where e =
= 0, A 1 A 2 =.0, A2+e- 2 *= 0,
cosh u cos a cos v
sin a
and therefore we can find A by the solution of equations of
Riccati's form.
SURFACES WITH ONLY PLANE LINES OF CURVATURE 125
Wo see that (cosh u cos ex cos v) A
= sin a sinh ui + sin a sin vj + (cos a cosh u cos v) k.
where i,j, k are iixed unit vectors at right angles, will satisfy
the conditions ; and we know that any other possible value of
the vector A can be obtained from this vector by a mere fixed
rotation.
The surface may therefore be regarded as the envelope of
the plane
x sin ex sinh u -f y sin a sin v 4- z (cos a cosh u cos v)'= U+ V.
(96. 11)
97. Enneper's theorem Let us now consider a curve
which is an asymptotic line on the surface.
We have y ? = for an asymptotic line and therefore
cos ft
P
If p is infinite the asymptotic line is straight and therefore
the surface is ruled.
Leaving aside the case of ruled surfaces, cos is zero and
therefore = , that is, the osculating plane of an asymptotic
Lt
line is a tangent plane to the surface.
For an asymptotic line the angular velocities are
1 1
^ r=0,
p* + (-j^ r\ (-^,, r\ =
and the formula
gives I 2+ _J_ 7> = 0) (97.1)
that is, the torsion is \/ K. This is Enneper's theorem.
We also see that the geodesic curvature of an asymptotic
line is just the ordinary curvature.
98. The method of moving axes. If we now return to
the equations of Codazzi (90 . 3), which are the foundation of a
considerable portion of Darboux's method of treating problems
126 CURVES IN EUCLIDEAN SPACE
of differential geometry, a method which is in effect the method
of moving axes, we may take ', " to be zero.
The rotations are p', q', r' ; p" t q", r", and the translations
'' 'n't Q > "> V? y an d the connexions are
q"r' t <l',-<l'\ = r'p"-r"p',
7's-V't - r'S"-r"t', !>' l" ~2>" l' - q't"-q"t'.
The displacements of a point whose coordinates with
reference to the moving axes are x, y, z are, with reference to
fixed axes with which the moving axes instantaneously
coincide,
dx + gdu + "dv y (r'du + r"dv) + z (q'du + q"dv),
dy + rj'du + rj"dv z (p'du+p"dv) + x (r'du + r"dv),
dz x(q'du + q"dv) -f y (p'dii +p"dv).
Thus for a curve on the surface making -an angle o> with
the axis of x
ds cos co = 'du+ "dv, dsama> = i]' du + r)" dv. (98 . 1)
A point on the normal to the surface and at unit distance
from the surface traces out what we call the spherical imago
of the surface.
Thus the spherical image of the curve is given by
da cos = q'du + q"dv, dcr sin 6 = p'dup"dv. (98 . 2)
The direction of the line element conjugate to the line
element whose direction is co is 6 -f - , and therefore the two
i
elements du, dv and Su, Sv will be conjugate if
?8v, + t"8v_T,'tu + j'8v ( .
2/du+p"dv ~ q'du + q"dv V ' ;
The asymptotic lines, being the lines traced out by self-
conjugate elements, will therefore be given by
dv
' ( '
p'du + p"dv
THE METHOD OF MOVING AXES 127
The spherical image of the surface will be given by
d(T 2 = (p'du + 1>" dv) z + (q'du + q"dv)*. (98 . 5)
The principal radii of curvature and the lines of curvature
will be deduced from the fact that the point whose coordinates
are R
will have no displacement in space and therefore
(' + Rq') du + d" + R<f) dv = 0,
It follows that the measure of curvature will be given by
rt/u"__/V r/ r "
Here we should notice that the translation functions depend
only on the ground form, as
e = (I') 2 + (I")', / = IV + !' V. .9 = (V) 2 + (O 2 ,
and that r' and ?" can be expressed in terms of the translation
functions, so that we see again and very simply that the
measure of curvature is an invariant.
If the surface is referred to parametric lines at right angles
we may take ( ; 6 2 __ A 2 du 2 + JPdv f2 9
and ' = A, " = 0, rj' = 0, r?" = B.
We then have r' = ~ , r" = ~ ,
ti A
and at once deduce the formula
If we refer the surface to the lines of curvature as para-
metric lines we have p r = q" = 0, and the principal radii of
curvature are A R
R'=-^ t , R" = ^' (98.8)
99. Orthogonal surfaces. To illustrate the employment
of moving axes depending on three parameters we might
consider the case of orthogonal surfaces t
u = constant, v = constant, w = constant,
128 CURVES IN EUCLIDEAN SPACE
and take as axes the normals to these three surfaces at
a point of intersection.
We have " = "' = 0, i/" = rf = 0, { = f" = ;
and we may write g ^" 7^ '" .
The equations satisfied by the translation functions now
become ^ = _/,, & = ^/, V; , = _^/', , 1 = r ",
fi = -^/'", ^ = 9X",
^,/' + ,r'" = 0, f /" + ft/ = 0, 9i / + ^/" = 0.
We therefore have
^' = 0, a " = 0, r'" = 0,
and we have the well-known theorem that the lines of
curvature on the surfaces are the lines where the orthogonal
surfaces intersect them.
We shall return to the theory of orthogonal surfaces later
and so shall not pursue the study further here.
CHAPTER VII
THE RULED SURFACE
100. Let a vector trace out any curve in space, and let
X', yn', i/ f be unit vectors drawn through its extremity, parallel
respectively to the tangent, principal normal and binomial
of the curve. Let
X == cos 6 .X' sin sin . p! -h sin cos </> . v \
p, = cos . fj. f + sin . i/',
p = -sin 0.X' cos sin . /*' + cos cos 0./, (100. 1)
then X, //, *> will also be unit vectors mutually at right angles.
On the unit sphere, whose centre is the origin, vectors
parallel to these two sets will cut out the vertices A, B, C
and X, F, Z of two spherical triangles as in the figure.
Let - and - denote respectively the curvature and torsion
P o-
of the curve and let
1 , sin d> , cos , , rt .
? / = 0+ , 9'= , r' = , (100.2)
<r ^) />
and ^) = p' cos 5 + r' sin ^, (/ = q' 0, r = r' cos fl p' sin 0.
(100.3)
2843 S
130 THE RULED SURFACE
By aid of Serret's equations we see that
\ = TfjL qv ) (Lpv r\, v = q\pn, (100.4)
so that p> </, r are the rotation functions for the moving
triangle XYZ. The dot above any symbol denotes that it
is the symbol differentiated with respect to the arc of the
curve traced out by : we denote the arc by v.
101. The ground form and fundamental magnitudes.*
Let z = +uA, so that u is the distance of the extremity of
the vector z from the extremity of the vector As u and v
vary, the vector z will trace out a ruled surface of the most
general kind if Q and are functions of v.
The curve traced out by will lie on the ruled surface : it
is called the directrix of the ruled surface. Any curve on
the surface may be taken as directrix.
We have
Z 1 = A, z. 2 = A' + u A cos \ + urp (sin + uq) v.
The ground form of the ruled surface will be
ds 2 = du* + 2 cos Qdudv + (u z (q* + r 2 ) + 2 u? sin 0+1) <t/ 2 .
(101. 1)
We may write ? = ]\f cos ^ r = j|/ s j u ^ ?
when the ground form becomes
efe 2 = du z + 2 cos 0rftt dv + (u 2 -J/ 4 + 2 u3/ sin cos ^ + 1 ) c/t' 2 ,
(101 .2)
so that q and r are given when the ground form is given.
The function h is given by
k 2 = u 2 JI/ 2 + 2uJlf sin cos ^ + sin 2 0,
= (uJf + sin cos ^/r) 2 + sin- sin 2 \/r. (101.3)
The angle between two neighbouring generators is
Mdv, (101.4)
and the shortest distance between them is
sin 0sin ^ dv. (101 . 5)
The unit vector whose direction is the shortest distance is
cos i/f p, + sin ^ v. (101 . 6)
* [Soo also 22.]
THE GROUND FOKM AND FUNDAMENTAL MAGNITUDES 131
Since z l = A, # 2 = cos 0X + uM sin-v/r/j (uM cost/r + sin d) j>,
we have r ? 2 = (uM cos \/r + sin. 6) JJL -f uJf sin ^/r j/.
The unit vector normal to the surface at the extremity of z
is therefore Z, where
Z = /6' 1 [(uJfcos>/r + sin5)/z + ^^/sin\/rj/]. (101 . 7)
If we calculate 3 n , 12 , s 22 we deduce, by aid of the formulae
J2 11 = z^Zj J2 12 = Z^L, J2 22 z^Z y
that /2 n = 0, /2 12 = /fW sin ^sin 0,
/2 22 = p/^ 4- /^ -1 ((it 2 J/ 2 + u3f cos *//- sin 6) r/r + uM sin *// sin
+ J/cos(9sin^(sin^-ufl)). (101 .8)
We may write N for M cos \/r sin when we are only con-
sidering the ground form.
102. Bonnet's theorem on applicable ruled surfaces.
We saw that one of the most difficult problems in the Theory
of Surfaces was, given the ground form, to determine the
surfaces in space to which the form was applicable ; and wo
saw that the solution of the problem depended on a partial
differential equation of the second order. In general we can-
not solve this equation, but there is a striking exception in
the case of the ruled surface.
Let us first consider a theorem on ruled surfaces.
If on the surface with the ground form a^dx^dx^ the
curves x 2 = constant are geodesies, we must have {112} = 0.
If the curves x 2 = constant are asymptotic lines we must
have /2 11 = 0. If both these conditions are fulfilled the
surface is ruled; that is, if
n n = and {112} = 0, . (102.1)
the surface is ruled and the generators are
<r 2 = constant. (102.2)
Now suppose that we have a second ruled surface with the
same ground form and therefore applicable on the first surface,
and suppose if possible that its generators are not the lines
X) = constant.
132 THE RULED SURFACE
We can therefore choose our coordinates so that the two
surfaces will have the same ground form and that in the first
surface oc 2 = constant will be the equation of the generators
and on the second surface x 1 = constant will be the equation
of the generators.
We have /2 n = and /2' 23 = 0,
and as for the two surfaces fl u fl^ n^
and /2' n /2' 22 -/r^,
are the same, we must have
/2* 2 =/2'?.>. (102.3)
From Codazzi's equation (48 . 4) for the two surfaces we have
It is therefore possible to satisfy Codazzi's equation for
the given ground form with
{112} = 0, {221} = 0, (102.4)
by taking fl u and /2 22 both zero : that is, it is possible to
find a surface with both systems of asymptotic lines straight
lines; that is, to find a quadric applicable to the given
ground form.
Unless then the form
du* + 2 cos Odu dv+(M 2 u* + 2Nu+l) civ* (102 . 5)
is applicable to a quadric, the generators of any ruled surface
which is applicable to it must be
v = constant. (102 . 6)
This is Bonnet's Theorem and Bianchi's proof of it.
When therefore the ground form is given in the form
d** = du* + 2 cos 0dudv+(M 2 u 2 + 2Nu+ 1) dv 2 , (102 . 5)
we know that, leaving aside the case of quadrics, the surfaces
which are ruled and applicable. on it must be generated in the
method we have described [so that their rectilinear generators
are applied to its rectilinear generators].
BONNETS THEOREM ON APPLICABLE RULED SURFACES 133
When the ground form is given we are given q and r. Wo
may take p as any arbitrary function of v. We then know
p and <r of the directrix, and so can find it by the solution of
Riccati's equation. Similarly we obtain X and thus find the
ruled surface.
103. Ground forms applicable on a ruled surface. If
we are given the ground form of a surface, how are we to
decide whether it is applicable on a ruled surface? It will
be applicable on such a surface if the ground form can be
brouht to the form
du* + 2 cos Odudv+ (M 2 u* + 2iVW 1) <lv 2 , (102 . 5)
where $, J/, and N are functions of v only, but unless these
are given functions of the parameter the general method will
not immediately apply. This is the question we now wish
to consider.
The expressions du , dv
r and -T-,
as as
where u and v are the parameters of a point on the surface,
are tensor components. We may denote them by T l and T l .
The difficulty of the tensor notation feomes in when we
want to express the power of a tensor component with an
upper integer. Thus the square of T 2 would have to be
written T 2 T 2 , and in calculations this is inconvenient.
We therefore generally write the above two components as
and ?/ and try just to remember that they are tensor com-
ponents when we apply the methods of the tensor calculus/
The equations of a geodesic are ( 38)
+ {112} f + 2{122J^+{222], / 2 = 0,
and <l-. f *t +n li
its C 5a; 1 'do--/
^ = *li + ) ,te*.
134 THE RULED SURFACE
The equations of a geodesic may therefore be written
+ 1121} +[221] ?=(),
(103.1)
Now these equations are very simply expressed in the
tensor notation by
T 1 T.\ + r^ = 0, T^T:\ + y-T.^ = 0. (103 . 2)
The equation of the asymptotic lines is given by
n ll T i x i + 2n u T l T*+n M 'i*T* = o. (103.3)
Now remembering that on a ruled surface one of the
asymptotic lines is a geodesic, and taking the tensor derivatives
of this equation, we have
T* (/2 la 2V, + 12, 2 T.\ ) = 0,
(103.4)
and /2 ll . 2 7 T1 2 Tl + 2/2 12 . 2 T 1 ? T2 + /2 2 ,. 2 7 T2 r
+ 2^(/2 11 TJ a +/2 12 ^ 2 ( + 2r^/2 12 IV 2 + /2 22 ^ 2 ) = 0.
(103.5)
Multiplying the first equation by T l and the second by T 2 ,
and adding, and making use of the equations for a geodesic,
we see that if the surface is ruled we have for the equations
of that asymptote which is a generator
<>, (103.6)
. 2 V i = 0. (103.7)
If we write these two equations
(a, b, c, dlg, rj) ?> = 0,
the eliminant is (Salmon, Uiylier Algebra, 198)
a*C J - 6 abBC* + G acC (2 B*-AC) + ad (6 ABO- 8
+ 9 b i2 ACr z - 18 6c JJ3C+ 66d4 (2 B*-AC) + 9 c
-fycdBAt + dtA*^ 0. (103.8)
This vanishes for a ruled surface.
GROUND FORMS APPLICABLE ON A RULED SURFACE 135
Now we know that
n n a,,-ni 2 =K(a n a^-ai 2 ) t
and, since an arbitrary function is needed to express /2 n , /2 22 ,
fl vj , in terms of the parameters, there can only be one other
eq nation connecting these functions.
Applying tensor derivation to this equation we have,
using Codazzi's equations,
(103.9)
Wo thus have three equations, viz. these two and the
oliminant we have found. We conclude that this system
must be complete if the surface is ruled. For if another
equation of the first order in the derivatives of /2 U and fi^
could be obtained the function /2 12 is known in terms of /2 n
and /2 22 wo could obtain /2 n and /2 22 by quadratures, and
no arbitrary function woulTl appear.
This method, though tedious actually to carry out, will
enable us to determine whether any given ground form is
applicable to a ruled surface.
104. Case of applicability to a quadric. We must now
consider the ground form
du 2 + 2 cos dudv -f (M 2 i<? + 2Nu + 1) cZi; 2 ,
as regards its special form when it is applicable to a quadric.
The Cartesian coordinates of any point on a fixed generator
of a quadric may be taken to be
av + b ' J av + b ' av + b A
where the variable v denotes distance on that generator.
We have similar expressions for the coordinates on any
other generator ; and the variables v and v' of the points
which lie on the same generator of the opposite system will
be connected by a bilinear equation.
It follows that if P l is a point on the first generator and P 2
136 THE KULED SURFACE
its correspondent on the second generator the direction cosines
of their join may be taken as
1) D D '
where a^ , b lt c l , <l } , a, 2 , 6 9 , 2 , d t> are linear functions of v and
jD 2 = dj (af 4 &1 +c|) +d| (a | -f 6J + c| )
-2c/ 1 rZ 2 (r/ 1 2 + 6 1 6 2 -h^ 1 ^). (104 3)
The coordinates of any point on the quadric may then be
expressed in terms of u and v in the form
61 (? 6.j ^i
- 1 -"
(104.4)
It follows that J/ 2 , iV/>, coaOD are rational functions of ?;
which can be calculated, and that J) 2 is a quartic in v.
105. Special ground forms. Binormals to a curve. Line
of striction. We have found in 100, 101 the chief formulae
required in the study of the general ruled surface. When
the ground form is given we are given q and r, and we find
the different ruled surfaces which are applicable to the form
by varying p. This generally means that we vary 0, the
angle of inclination of the osculating plane of the directrix
to the corresponding normal section of the surface. We
cannot however take (/> to be zero unless the ground form is
special : for, if is zero, q -f- 6 is zero : that is,
M cos \ls + = 0;
which would give the special ground form
cfc 2 =i <lu* + 2 cos Odudv + (M 2 u 2 - 2 6 sin 0u -f 1 ) dv*.
(105.1)
Thus the binomials to a curve in space trace out a ruled
surface with the special ground form
d** = du* -f ~ + lrf v 2, (105 . 2)
where cr is the radius of Wsion.
THE LINE OF STRICTION , 137
If we take as directrix an orthogonal trajectory of the
generators, is , and the ground form is
2
d8 z = cZu 2 + (4/ 2 u 2 + 2 Mu cos ^ + 1 ) cfo 2 . * (105.3)
In seeking the surfaces which are ruled and applicable to
this form we may take for one of them cf> = i . The directrix
2
of this surface will be an asymptotic line and the surface will
be generated by the principal normals of this directrix.
We obtain the equation of the line of striction from the
ground form itself. We have to find for given values of v
and dv the values of u and dw which will make
du? + 2 cos Odudv + (JfcPu 2 + 2 M n cos ty sin 6 4- 1) dv 2
least.
Clearly we must have
dtb + cosOdv 0,
3/K, -f cos \//- sin = 0.
The equation of the line of strictiou is therefore
J/u + cos^sinfl = 0. (105 .4)
Let us take the line of striction as the directrix. We must
then have cos i/r sin = 0. (105.5)
We cannot have sin 6 equal to zero unless the shortest
distance between neighbouring generators vanishes : that is,
unless the ruled surface is a developable. We must therefore
have in general, when the line of striction is taken as the
directrix, ^ = - , and therefore q = 0.
2
It follows that 6 = , i.e. that $, the rate of increase of
P
the angle at which the line of striction crosses the generators,
is equal to the geodesic curvature of the line of striction. It
follows that the line of striction will cross the generators at
a constant angle if, and only if, it is a geodesic. In this case
the ground form will be
tte 2 = du 2 + 2 cos 0(dudv 4- (Jf ?u 2 -f 1) dv*, (105.6)
where a is the constant angle of crossing.
138 THE RULED SURFACE
7T
If a = - the form will be applicable on the surface gener-
ated by the binormals of a curve in space.
106. Constancy of anharmonic ratios. Applicable ruled
surfaces and surfaces of Bevolution. We shall now con-
sider the equation of the asymptotic line which is not
a generator.
The equation is 2f2 n du + f2 22 dv = 0. (106 . 1)
Referring to the values given for /2 12 and /2 22 wo see that
this is an equation of Riccati's form. It follows that the
equation of an asymptotic line is
<">
where a, /?, y, S are some functions of v only, and k is an
arbitrary constant.
We thus see that every generator is cut in a constant
anharmonic ratio by any four fixed asymptotic lines.
We also notice from the property of Riccati's equation that
if we are given any one asymptotic line we can find the
others by quadrature.
We have also seen in 101 that the normal to the ruled
surface is parallel to
uH(cos tyfj, + sin tyv) + sin dp.
It follows that the anharmonic ratio of four tangent planes
through any generator is
(Ut-uJXut-uJ
- 9 v '
that is, the anharmonic ratio of the planes is the same as that
of the points of contact.
Suppose now that P is any point on a generator, and that
the tangent plane at P intersects a neighbouring generator
in P'. Then in the limit PP' is the element of the asymptotic
line at P. It follows that the asymptotic lines through four
points on a generator intersect a neighbouring generator in
ANIIARMONIC PROPERTIES 139
the cross ratio of the tangent planes : that is, in the cross
ratio of the points of contact.
We thus have a second and more geometrical proof of the
theorem that every generator is cut in a constant cross ratio
by four fixed asymptotic lines. This theorem also is duo
to Bonnet.
The condition that the normals to a ruled surface, at two
points u l9 u 2 on the same generator, may be perpendicular is
u^u^M** (M! + w- 2 ) Msin <9cos i/r + sin 2 = 0. (106 . 4)
The points are therefore corresponding points in an in-
volution range whose centre is on the line of striction.
No ruled surface exists which is also a surface of revolution
except the quadric of revolution. We see this at once by
considering a surface of revolution in relation to any meridian
line. The asymptotic lines, through any point on this line,
must be symmetrically placed with respect to the line. If
then one of these is a straight line so will the other be. The
surface will therefore, if it is a ruled one, be a quadric.
But the ruled surface may be applicable on a surface of
revolution without being a surface of revolution. We now
inquire what property the ground form must have if it is to
be applicable to a surface of revolution with generators
corresponding to the meridian lines.
Taking as directrix an orthogonal trajectory of the genera-
tors we have e ^a = <J U 2 + (j; 2^2 + 2 tf u + i) <l v * m (1 OG . 5)
If then this form is to be applicable to a surface of revolu-
tion M and JVmust be constants, and we see that the ground
form may be written
rfs 2 = tin* + (u- + a 2 ) dv* ^ ( 1 06 . G)
where a is a constant.
Thus the catenoid and the helicoid will both have this
form applicable to them.
107. Surfaces cutting at one angle all along a generator.
We now wish to investigate the cor^dition that two ruled
surfaces with a common generator may intersect at the same
HO THE RULED SURFACE
angle all along that generator. The condition will be found
to have an interesting connexion with a particular class of
congruences.
We have seen that
uM (cos \ITJJL + sin -fyv) -f sin 0/j,
is a vector parallel to the normal to the ruled surface at the
extremity of the vector 0.
As we move along the generator this vector turns through
an angle, remaining of course perpendicular to the generator.
The vector product of the above vector and the neighbouring
vector ( u + d u ) M (cos i//yz 4- sin tyf) -h sin 6/n
is Mdiis'm 6 sin tyX.
But the vector product is also
(M 2 u* + 2 MM cos ^ sin + sin 2 0) </eA,
where de is the angle turned through ; and therefore
^ _ _ -Wain SUM/T . (1071)
du M-u* + 2 Ala sin flcos^-f H\I\* ^ ' '
Let M k sin sin \r
where k is the ratio of the angle between two neighbouring
generators to the shortest distance between them. Then
dti (ku + cot ^)~ + 1
The equation of the line of striction is
ku + cot \fs = 0.
If therefore we measure u not from the directrix but from
the line of striction we have the formula
It follows that if we have two ruled surfaces, for which k
is the same, and one of the surfaces is given a movement in
space, bringing one of its generators into coincidence with
CUTTING AT ONE ANGLE ALONG A GENERATOR 141
the corresponding generator of the other, and the correspond-
ing points of the line of striction into coincidence, then the
two ruled surfaces will intersect at the same angle all along
that generator.
108. The ruled surfaces of an isotropic congruence.
Let us now consider a ruled surface referred to its line of
striction as directrix.
Now ( 100) ' = A' = cos <9A-sin 6v.
and, since the line of striction is the directrix, q is zero. We
therefore have /A = 0. (108 . 1)
Suppose now that is a vector depending on two para-
meters u and v, and that A is a unit vector depending on the
same two parameters.
Consider the congruence z + w\.
The congruence is said to be isotropic if and A correspond
orthogonally. [See 81.]
We have as the conditions for orthogonal correspondence
<L A . = > <L^ + ^ 1 = ' Q*=>
and therefore Aj = A^, A 2 = c/A^ 2 ,
where a is some scalar function of u and v.
We thus have d\ - a\d, (108 . 2)
whatever be the values of du and dv.
The ruled surfaces of the congruence are obtained by con-
necting u and v by some equation. For any ruled 'surface of
the congruence wo therefore have
f\ - 0, A = aA^
where the dot denotes differentiation along the arc of the
curve chosen. This arc will be the line of striction since
142 THE RULED SURFACE
and, since q is therefore zero,
X = r cosec 0A
It follows that a = r cosec 0,
that is, from our definition of k ( 107),
a = L (108.3)
The ruled surfaces of the isotropic congruence therefore
intersect at the same angle all along their common generator.
They have all the same k at the point where the common
generator intersects the surface w = 0, and their lines of
striction all lie on this surface. This surface is the central
surface of the congruence.
CHAPTER VIII
THE MINIMAL SURFACE
109. Formulae and a characteristic property. If we
give to z y the vector which traces out any surface, a small
arbitrary displacement normal to the surface at the extremity
of 2, we have z' = z -f A , where is a small arbitrary parameter.
Since z'^Zi + W + Xtl, s' a = 3 2
we have
a' u = a n 2z l \ 1 t ) a' ]2 = a 12 22 1 A 2 , 22
that is, by (50 . 9),
(109. 1)
If the area of the surface is to be stationary, under this
variation, then a must be stationary, where
a = a 11 tf 22 -ai 2 ,
r
since the area is a^dudv.
j
We therefore have
an^22 + 22'ii- 2a iAa = 5 (109.2)
that is, the sum of the principal radii of curvature must
be zero.
The surface of minimum area, the minimal surface as it is
called, is therefore characterized by the property
22 / + Ji" = (109.3)
where R' and R" are the principal radii of curvature.
If we refer to lines of curvature as parametric lines on any
surface *j = .R'A lf z 2 =#" t A 2 ,
and therefore, if cd
144 THE MINIMAL SURFACE
is the ground form of the surface, the ground form of the
spherical image will be
<*udu* a Z2 dv* ( .
WF (^")*
The surface and its spherical image will therefore be similar
at corresponding points if, and only if,
(R")\ (109.5)
that is, if the surface is a sphere or a minimal surface.
On a minimal surface
and therefore 2 'A 12 + R'^ -f R\\ 2 = 0.
It follows that
and therefore without loss of generality we njay say
jR'Af=-l, Jf2 / Xi=-l. (109.6)
The ground form may then bo taken as
R'(du 2 + dv 2 ), (109.7)
the asymptotic lines as d f u? dv 2 = 0, (109 . 8)
and the ground form of the spherical image as
(R')- l (du* + dv*). (109.9)
Wo may now write R instead of R' t and since the ground
form of a sphere of unit radius is
we must have R (d 2 + sin 2 Od<f> z ) = dv? + dv 2 . (1 09 . 1 0)
A
If we take w = cot-e 1 ^,
2
we see that the complex variable w is the complex variable
on the plane on to which the sphere of unit radius can be
projected stereographically from the pole, if we take the pole
as the origin from which 6 is measured and take the plane as
the corresponding equator. *
FORMULAE AND A CHARACTERISTIC PROPERTY 145
If w denotes the conjugate complex
A
we see that 4 sin 4 diu dw d& 1 -f sin 2
2
A
and therefore 4 R sin 4 - div dw = cZu 2 -f (v 2 . (109. 11)
If wo regard u + tv as the complex variable of another
plane and denote it by x, we have
1
4
Now the curvature of the form
4 jfisiri 4 dvnlw
4
= cosec*
is zero ; and, from the formula for the curvature of the
ground form ( faa 2fdudv,
we have f*K =/ lt / a ff l2 ,
and therefore ft sin 4 =f(w)F(<iv), (109. 12)
Z
where / and F are functional forms. If the surface is to be
a real surface these forms must be conjugate forms.
A
Since cosec 2 1 -h ww
the formula for R may be written
R = ( 1+ ivwff (w) F (w). (109.13)
We notice that in a minimal surface the asymptotic lines
are perpendicular to one another in general though not
necessarily so at a singular point. This property is character-
istic of the minimal surface.
110. Keference to null lines. Stereographio projection.
We now choose as the parametric lines on the minimal
surface its null lines ( 45) and, instead' 1 of writing iv and w,
we take u and v to represent tAese complex quantities.
2843 U
146 THE MINIMAL SURFACE
"The spherical image will therefore also be referred to ifes
null, lines and their parameters will be the same w and w or
u and v.
The normal to the surface is therefore given by
(1 + uv) X = (u + v)t'-i(u-v)i"+(uv-l)i f ", * (110.1)
where /, t", i" are three fixed unit vectors mutually at right
angles and i denotes V 1.
It is now convenient to introduce two vectors defined by
. (110.2)
These vectors are conjugate vectors and of course not real.
They are, in fact, generators of the point sphere whose centre
is the origin.
Such point spheres must play in solid geometry the same
part that the circular lines through a point play in plane
geometry. We may easily verify the following relations
between p, a, and X ;
= t(\ + uv) <2 X,
2p = (l+iiv)*\ 2 , 2(7 =
P\ = ip, CrX = i(T,
pA x = \, crX 2 = ~fX,
-2(X, P 2 = 0, cr 2 = 0. (110.3)
We have seen ( 109) that the complex variable u on the
sphere Q
u = cot-e </> (110.4)
is the complex variable on the equator when we take the
ground form of the sphere to be
and project the sphere, from the pole from which we measure
d, stereographicaJly on to the equator.
The conjugate corrfplpx v is the image of u in the real axis
of the plane. <
REFERENCE TO tfULL LINES 14?
The complex u fixes a real point on the sphere, since wh&ri
u is given its two parts are given and so its conjugate v is
given. If Uj is the complex which fixes a point I\ on the
sphere and u 2 is the complex which fixes the diametrically
opposite point on the sphere, we have
1+1^2 = 0, (110.5)
and consequently we also have
We should notice that we cannot have
uv +1 = 0.
The complexes which correspond to the two opposite ends
of a diameter may be called inverse complexes.
111. The vector of a null curve. A null curve is defined
as a curve whose tangent at every point intersects the circle
at infinity. Another way of stating the same definition is to
say that the tangent at every point is a generator of the point
sphere at the point. If z is the vector which traces out
a null curve we therefore have
Now the components of a vector which satisfies the equation
X" = may be taken as proportional to
and therefore we must have
where/" (u) is some scalar function of the parameter u.
It follows that the vector z which traces out a null curve
may be defined by
since the third derivative of p vanishes.
We now denote the vector of the null curve by a, where
a = pf" (u)- Pl /(u) + p u /(u). (111.3)
112. Self-conjugate null curves. They may be (1) uni-
cursal, (2) algebraic. The conjugate nujl curve to a is clearly
(112.1)
148 THE MINIMAL SURFACE
where / is the conjugate function to /, and a- the vector we
have defined in terms of its parameter v.
A null curve is said to be self-conjugate, when for each
value of u a value v' can be found, where v' is the conjugate
complex to a complex u', such that
We generally write p without specifying its parameter u,
but sometimes we may need to bring the parameter into
evidence and then we write it p u .
Differentiating the equation
<*u = <V
we have pj'" (u) = <v/'" (v 9 )^,
so that Vpu<r v f = >
and therefore 1 + u v' = 0. (112.3)
If we now write p for p u , and o- for ay, we "have
p + 0u 2 = 0, PJ + 0-2+20-U = 0, tt 2 p n + (T 22 + 2ii(r 2 + 2u 2 (r = 0,
and we can write
= - <7U 2 /" () + K + 2<TW.)/' (It)
-(<r. 2 + 2w<r 2 +2uV)^, (112.4)
tv
^ = -/" ("') - ^/' (^) + ^ 2 /(^'). (112.5)
If we now equate the coefficients of the vector cr^ on the
two sides of the equation a =a, /
we see that /(u) = - it 2 /( --- ) (112.6)
and we see further that this single condition is sufficient to
satisfy the equation # . ^ lt (112 7)
tt
In order then that a null curve may be self-conjugate it is
necessary and sufficient that the function / which defines
it should have the property
(112.8)
SELF- CON JUG ATE NULL CURVES 149
If we take
7 > = >-
(112.9)
where the coefficients are any real constants and the summa-
tions may pass to any limits, we see that the function will
satibfy the condition necessary to determine a self- conjugate
null curve ; and we see that this is the most general function
which will do so.
If we only take a finite number of constants the self-
conjugate null curve which results will be unicursal,
More generally, if we take /(n) to be an algebraic function
of u y then /' (u) and /" (u) will also be algebraic functions
of u. We can then express the Cartesian coordinates of any
point on the self- conjugate null curve rationally in terms of
f( u )> /' ( u )> f" ( u ), and u. Wo shall then have six algebraic
equations, connecting the three Cartesian coordinates and the
four quantities /(u), /' (u), f" (u), and u. We can eliminate
these four quantities and there will result two algebraic
equations connecting the Cartesian coordinates.
We have now seen how to construct null curves and self-
conjugate null curves; and also how we can construct
sell -conjugate null curves which will bo unicursal; and yet
more generally how to construct self conjugate null curves
which will be algebraic.
113. Generation of minimal surfaces irom null curves.
Double minimal surfaces. When the minimal surface is
referred to the null lines on it as parametric lines we have
n = 0> o 22 = 0,
and therefore, since a 11 /2 2:J -f-a 22 /} 11 = 2# 12 /2 12 , (113.1)
we must have /2 12 equal to zei'o.
150 THE MINIMAL SURFACE
That is, we have
z\ = 0, z\ 0, z } A 2 = ^Aj = :
and therefore, since \z l = 0, X0 2 = 0,
we have Xz l2 = 0.
We also have, from z\ cu = 0,
that -1^12 = ^ ^2^12 == 0>
and therefore 12 pA*
where > is a scalar. But A: 12 = 0,
and therefore z l2 = 0. (113.2)
The minimal surface is therefore a particular case of a
translation surface.
A translation surface is defined by
s = a + ft * (113.3)
where a is a vector describing a curve whoso parameter is u
and ft a vector describing a curve whose parameter is v. We
see why it is called a translation surface as we can generate
it by translating the u curve along the v curve or translating
the v curve along the u curve.
We might also define a translation surface by
20 = a + /3, (113.4)
when we see that it is the locus of the middle points of
chords one extremity of which lies on one curve and one
on the other.
In the case of the minimal surface we also have
(cZa) 2 = and (d/3) 2 = 0,
since z\ = and z\ = 0.
The minimal surface is therefore given by
where a and /3 are vectors tracing out null curves.
If we confine ourselves to ral minimal surfaces the null
GENERATION OF MINIMAL SURFACES 151
curves must be conjugate and the parameters of the two
points must be conjugate complexes. It is obvious that such
conjugate null curves will, if the corresponding parameters
are conjugate complexes, give a real surface, and the converse
may be proved.
If the null curve is a self- conjugate curve, however, we
must take as the corresponding complex, not the conjugate
complex, but the inverse complex.
Thus the general real minimal surface is given by
20 = <x IL + a v ] (113.5)
and the real minimal surface generated from a self-conjugate
null curve is given by 2s = a M +a/_i\, (113.6)
\ ~ ?/
where the suffix is the parameter of the null curve which is
to be taken.
We notice that in the minimal surface
1Z = + /_ IN,
as we pass from the point whose parameters are u, v by
a continuous path to a point whose parameters are ,
we return to the point from which we started ; the z of the
point will be the same but the X will be changed into X.
That is, we are on the other side of the surface. For this
reason the surface is called a double minimal surface.
114. Henneberg's surface. We have now seen how
minimal surfaces are generated from null curves, and how real
minimal surfaces are to be obtained, and how i;eal double
minimal surfaces may be generated.
From what we said about the construction of null curves
we see how to obtain minimal surfaces which will be rational
functions of their parameters and how to obtain more generally
algebraic minimal surfaces; and from what we said about the
construction of self-conjugate null <jui*ves we can construct
these surfaces to be double minimal surfaces.
152 THE MINIMAL SURFACE
Thus
will be an example of a real double minimal surface as may
easily be verified. It is known as Henneberg's surface.
It may easily be shown that a minimal surface will then
only be algebraic when the null curves which generate it
are algebraic.
115. Lmes of curvature and asymptotic lines on minimal
surfaces. We have for a minimal surface
and, if the surface is to be real,
2s =
It follows that
and therefore 4 -^ = /'" (it)/"' (v) pJr.
But 4/^ = (l+'uyJ'ATX,,
and, if R, Jl are the principal radii of curvature of the
surface, ^ _ ,> 2 r\
^1^2 ~ ~^ A 1 A 2'
We therefore have
U3R*-=f"(u,)f'"(v)(l+uu)*. (115.1)
then 4^=(l 4 t,),
, o
and 2s =
THE LINES OF CUKVATUKE AND ASYMPTOTIC LINES 153
We then have
so that if i//- ~ f- irj, \fr ?;
we come back to the ground form
ds* = R(d* + dri*)
for the surface.
The lines of curvature are
= constant, ?/ = constant,
and the asymptotic lines are
-f 77 constant, g rj = constant.
We may therefore say, if R(f>(u) denotes the real part of
(it), that one family of the linos of curvature is
R f Vf" (u) du = constant, (115.2)
j
r __
and the other Ri </f" (u)du constant ; (115.3)
j
whilst the asymptotic lines are given by
R f V7f tTr (^) du = constant, (115.4)
R (V-/'"(tf)<Ztt = constant. (115.5)
110. Associate and adjoint minimal surfaces. The surface
obtained by substituting for / the function e l( *f wjiorc a is
a real constant is said to be an associate minimal surface
to /; and when we take for a the number ~ it is said to be
the adjoint minimal surface.
An associate minimal surface is applicable on tho surface
to which it is associate and the gormals are parallel at
corresponding points.
154 THE MINIMAL SURFACE
If { is the vector which traces out the adjoint surface to z
2<b :-/'" (U)pdtl+f" (V) (Tttv,
2df = L (/'" (u) pdu-f" (v) <rdv), (1 16 . 1)
so that these two surfaces will also correspond orthogonally.
We see that z i traces out not a surface but a null curve,
and z + i traces out the conjugate null curve.
Since p\ //>, crA JCT,
we also see that d{ = Xth. (116.2)
If then we are given a curve on the surface we shall know
the which will correspond to z along this curve, if we
know the normal to the surface along the curve. We shall
therefore know c-f/^ and z i along the given curve, and
thus have the null curves which generate the minimal surface.
p ^ ^
The formula Oi (l -^-i\ \dz % (116.3)
j
is due to Schwartz.
CHAPTER IX
THE PROBLEM OF PLATEAU AND CONFORMAL
REPRESENTATION
117. The minimal surface with a given closed boundary.
Any account of minimal surfaces would be incomplete without
some reference to the problem proposed by Lagrange : ' To
determine the minimal surface with a given closed boundary,
and with no singularity on the surface within the boundary/
This problem is known as the Problem of Plateau, who solved
it experimentally. The problem has not yet been solved
mathematically in its general form ; but has been solved in
some particular cases, where the bounding curve consists of
straight lines and plane arcs of curves.
Consider a part of the bounding curve, which is a straight
line, on a minimal surface. This line must be an asymptotic
line on the surface. Now we saw ( 109) that, when the
surface is referred to the lines of curvature, as parametric
lines, the equation of the asymptotic lines is
dtf-dv* = 0; 017.1)
and the ground form of the surface is
R(<lu* + dv*)> (117.2)
and the ground form of the spherical image is
R- l (d>u? + dv*). (117.3)
Wo conclude that when the surface is conforinally repre-
sented on the plane, on which u and v are the rectangular
coordinates, the asymptotic lines are conformally represented
by lines parallel to the bisectors of the angle between the
axes, and the lines of curvature, and also their spherical
images, are conformally represented by lines parallel to
the axes. *
156 PLATEAU AND CONFORMAL REPRESENTATION
If a part of the bounding curve is a plane curve, whose
plane cuts the minimal surface orthogonally, and is therefore
a geodesic, it must be a line of curvature. It will therefore
be conforrnally represented on the plane by a line parallel to
one of the axes.
If then the whole of the bounding curve is composed of
straight lines and such curves, the bounding curve will be
conform ally represented on the plane by a figure, bounded
by straight lines, parallel either to the axes or to the
bisectors ; and the part of the minimal surface, within the
boundary, will be represented by the area of the plane within
the polygon.
Next let us consider the spherical image of the surface
within and on the boundary. At each point of the boundary,
the normal to the surface will be perpendicular to a direction,
which will not change as we pass along a continuous part of
the boundary, but will change at each angle of the boundary.
The boundary will therefore consist of * arcs of great
circles.
If therefore we can find a function of ^v 9 the complex
variable which defines the position of any point on the sphere,
which will transfoim the spherical boundary into the plane
boundary, and points within the spherical boundary to points
within the plane boundary, we shall have u-t iv known in
terms of w, and can proceed to find the required surface as
follows.
We have (109 . 10) for an element of the sphere
da* = dff* + sin 2 6d$\ (U 7 . 4)
so that w = cotf^ (117.5)
j
is the complex variable which defines the position of points
on the sphere.
The normal, to the sphere, which is given by w, is by (110 . 1)
(l+ww)\ = (w + w)i' i(w w)i" + (ww-~l)i" f , (117.6)
where i' t i"> i" are* fixed unit vectors, mutually at right
angles, MS \/ 1, and w the conjugate complex to w.
MINIMAL SUEFACE WITH GIVEN BOUNDARY 157
Now we know that in terms of u and v
and therefore R~ l (du? + dv*) = dO* + *iri 2 6d$*. (117.7)
As we have seen in 109, R is therefore known, being
given by
dx
(117.8)
dw
We can therefore construct the surface since R and A are
known in terms of w and w .
We can retrace our steps and see that the surface we have
obtained satisfies the conditions required.
We are thus led to the problem of con formal representation,
and this we proceed to discuss, so far as it bears on the
question before us.
118. The notation of a linear differential equation of
the second order with three singularities. Let a, c, b be
three real quantities in ascending order of magnitude, and let
x be a complex variable.
When x lies on the real axis between oo and a, or
between b and + oo , we see that
b G j' a
b a x c
lies between zero and positive unity. When x lies between a
and c c b x a
c ct x - b
lies between zero and positive unity. We also see that the
reciprocals of these two expressions lie between zero and
positive unity, when x lies between c and b.
When x is complex we see that the modulus of one of the
first two expressions is less than unity, or the modulus of
each of the reciprocals is less than unity.
Let a lf Of 2 , /?!, j8 2 , y 19 y a be six quantities real or complex,
but such that ^ + ^ + ^ + ^4.^ + ^ = i ; (118.1)
and such that the real parts of
a 2 - a i> ft-01. ,72-71
are each positive.
158 PLATEAU AND CONFORMAL REPRESENTATION
Let
a? a
(za)(x-l) (x c)Q
z+'-y'-y*. (1,8.2)
b x c
T
- - 1 - - j
x a xb x c
(118.3)
and let (a t , ft, y u 2 , ft, y a , a, 6, c, a)
denote the hypergeometric series
where p^^ + ft + yx, ? = l + ft + y l , r =
6 ic a
__
*
a x b
We notice that P and Q are unaltered by the following
substitutions :
focg, (ft/U ( yi y a ),
(yaai)(y 2 a 2 )M, (,^i) (*&)(&) (118.5)
110. Conformal representation on a triangular area.
Consider now the diffurential equation
It is known, and may easily be proved, that
is a power series, beginning with (x a)* 1 for its first term
and expansible in powers of x a in the neighbourhood of
x = a, which will satisfy the differential equation. This
series when x lies on the real axis is valid when x lies
between a and c. It is therefore valid at any point in the
plane, the circle through which, having a and 6 as limiting
points, intersects the real axis between a and c.
CONFORMAL REPRESENTATION ON A TRIANGULAR AREA 159
Another power series also beginning with (x a)* 1 can be
obtained from the first by applying the substitution
to it. The two series will therefore be identical at any point
where they are both valid. The second is valid for real
values of x between oo and a arid between b and -f oo.
The region for which it is valid, when x is complex, can be
obtained by a similar rule to that which was used as regards
the first series.
When one series is valid, but not the other, the valid series
is a continuation of the other. We denote these series by Fa r
l>y applying the substitution (oqaj we obtain two other
series beginning with (x a)S valid over the same part of
the plane. We denote these series by Fo^.
By applying the substitution (y^) (y 2 J (cu) we get two
series, Yy l beginning with (x c) 71 , and Fy 2 beginning with
(x c') 7z , valid over the part of the plane which corresponds
to real values of x between c and b.
By applying the substitution (Piyi)(P 2 y 2 ) (be) to these last
two series we get two other series, F/Jj beginning with
(x 6)^, and F/8 2 beginning with (x b)^ 2 , valid over the
same part of the plane. All these series, when valid, satisfy
the equation.
r . Ya 2
Let w -fr-^
Y*i
Then we see that, as x describes the real axis from 6 to -f oo ,
and then from oo to a, w varies continuously and its argu-
ment is TT (a 2 (Xj), if we agree that the argument of a positive
quantity is to be taken as zero, and the argument of a
negative quantity as ?r, as x describes the real axis in this
definite way. t *
Let Q be the point in the'iu plane which corresponds to
160 PLATEAU AND CONFORMAL REPRESENTATION
x = 6, and let P be the origin in the w plane corresponding
to x = a. As x describes the path defined, w describes the
straight line QP.
When x describes the semicircle about (6 the argument of w
diminishes by 7r(a 2 o^), and as x describes the real part of
the axis the argument of w remains zero, till we come to R,
which corresponds to x = c.
Wo must now consider what happens as x describes the
semicircle round c, and then, passing along the real axis,
comes to b and passes round the semicircle there.
Over any part of the plane which corresponds to real values
of x between c and b we can express w in either of the forms
C + D-
where A t B, C y D and A', B', (7, D' are certain constants.
We see this from the known properties of a linear differential
equation of the second order.
Now the argument of ~^ is the same as that of
* Yi
/X C, A7a-7i
( -(c-a))
\X--(t '/
and therefore zero, as x passes iflong the real axis from c to b.
CONFORMAL REPRESENTATION ON A TRIANGULAR AREA 161
It follows that w describes a circle which passes through
Q and E.
^
The increment of the argument of w ^ as we pass along
C
the semicircle c is the same as the increment of the argument
of y^ ; that is, it diminishes by ^(y^ yi)- The circular
arc through R therefore makes an angle 7r(y 2 "~yi) with jRP.
In the same way we see that the angle at Q is 7r(/3 2 jSJ.
Since, when x moves from its real axis to the positive side
of its plane, w must move to the inner part of the triangle
PQR y we see that the positive part of the plane of x is con-
formally represented by the inner part of the triangle.
12O. The w-plane or part of it covered with curvilinear
triangles. Consider now the transformation
x , ^px + q
rx + s
where p, q, r, 8 are any constants, real or complex.
If x describes a circle (or as a particular case a straight
line) in its plane, so will x'. If x l and x^ are any two points
inverse to the circle #, then x\ and o/ 2 will be inverse to
the circle x'.
We thus see that if P, Q, R are the three points which, in
the above transformation, with F 2 ~ Fo^ substituted for x,
correspond to the singularities at a, 6, c, the curvilinear
triangle PQR, formed by three circular arcs intersecting at
angles ATT, /i?r, VTT, where *
A = ,,-!, f= 0S-&, '"=72-71.
2843 y
162 PLATEAU AND CONFOKMAL REPRESENTATION
will enclose the part of the w plane, which conformally repre-
sents the upper part of the x plane.
Let w be the complex variable which defines any point 8
within tbe triangle PQR, and let w l be the complex variable
which defines the point 8 19 which is inverse to $ with respect
to the arc RQ.
Let rf
be the substitution which transforms the arc RQ to a part of
the real axis of w in its plane.
Then and (120.2)
rw + s m\ + s '
arc inverse to one another with respect to the real axis of iv.
Let f(x) be the function of x which we found would in
this case transform the upper part of the x plane to within
the curvilinear triangle in the w plane. We now assume the
quantities ot 19 a 2 , J3 19 /? 2 , y T , y 2 to be all real. Along the real
axis of the plane x the coefficients in /(*') will be real, and
therefore f(x) will be the function which will transform the
lower part of the plane x to points without the curvilinear
triangle, where x denotes the conjugate variable to x.
We therefore have
It follows that
w = - and w = - . (1 20 . 3 )
p-rf(x) l p-rj(x) 9 v ;
and consequently we have
p-rf(x)
Eliminating f(x) we have
* (ps qr)w + 08-^ qs
W =! * / _ 1_.
( jjw rp) w + p$ qr
A DERIVED NET- WORK OF TRIANGLES 163
If then w = F(x),
and w l </) (x),
then (x) = (fo-g*)g(g) + g'-gg . (120 . 5)
v '
If JFJ is the inverse of P in the arc Q-R, we thus see how
the lower part of the x plane is conformally represented on
the triangle P$R in the w plane.
Similarly if (^ is the inverse of Q in RP, and 7? x the
inverse of R in PQ, we can conformally represent the lower
part of the x plane on the triangle ^.RP, and on the triangle
R.PQ.
Just in the same way from the triangle 1\QR we can by
inversion obtain three other triangles, one of which will be
the triangle PQR. These triangles will give conformal repre-
sentations of the upper part of the plane x on the plane of w.
Proceeding thus we cover the whole, or a part, of the
^u plane with curvilinear triangles.
121. Consideration of the case when triangles do not
overlap. In general these triangles will overlap, so that
a point in the w plane may be counted many times over : in
fact, unless A, //, v are commensurable, a point in the w plane
which lies within any triangle will lie within an infinite
number of triangles. If, however, X, /z, v are each the reciprocal
of a whole number there will be no overlapping at all. We
now confine ourselves to this <!ase.
164
PLATEAU AND CONFORMAL REPRESENTATION
One and only one circle can be drawn to cut orthogonally
the arcs of the fundamental curvilinear triangle in the w
plane. By inversion we may take PQ and PR to be straight
lines.
We see that the two straight lines and the circle divide the
w plane into eight parts. We see, however, by considering
the original figure with which we began this discussion, that
the triangle with which we are concerned is the shaded one.
For at the point L the variable w will move in the direction
of the arrow, for a corresponding movement of z to the upper
part of the x plane ; and, as w will not move oft' to infinity,
the triangle could not be the outward part of
The triangle PQll is therefore of one of the two forms
In case (1) P must lie within the circle of which EQ is
the arc.
THE CASE WHEN TRIANGLES DO NOT OVERLAP 165
For, otherwise, the sum of the angles at Q and R being for
we are now assuming A = - , /* = -, ^ = ^
/I 1\
\q r/'
the sum of the angles at Q' and R' would be
,(2-1-1);
\ q r/
and therefore 2 < 1.
q r
But this is not possible if q and r are integers. No real
circle can therefore be drawn with P as centre to cut the arc
QR orthogonally in case (1).
The two cases are therefore thus distinguished : in case (1)
A + /* + i/>l, (121.1)
and the orthogonal circle is imaginary : in case (2)
X + ^ + ,/< l, (121.2)
and the circle which is orthogonal to the three arcs is real.
122. Case of a real orthogonal circle as natural boundary.
Taking case (2), the circle, whose centre is at P and which
cuts the arc QR orthogonally, must intersect the circle QR at
the points of contact of tangents to the circle from P. Clearly
these points are without the arc QR, since the arc QR is
convex with respect to P. The points P> Q, R therefore lie
within the orthogonal circle. When we invert with respect
to a point outside the orthogonal circle we have three circular
arcs within the new orthogonal circle*. By considering the
point P l which is the inverse bf P with respect to QR, we see
166 PLATEAU AND CONFORM AL REPRESENTATION
that P l also lies within the orthogonal circle. Proceeding
thus we see that all the curvilinear triangles are within the
real orthogonal circle which corresponds to the case
A+ fJL + V < 1.
In this case, therefore, only the part of the w plane which
lies within the orthogonal circle is covered with the curvilinear
triangles, which conformally represent the x plane on the
w plane. This circle is therefore the natural boundary of
the function which, with its various continuations across the
real axis of x, conformally transforms the x plane to the
w plane.
Since there are an infinite number of solutions of the in-
equality 1 i i
- + + - < 1,
P 9 r
where p, q, and r are integers, we get an infinite number of
triangles which grow smaller and smaller as we continue to
invert and invert : and as we approach the -boundary the
orthogonal circle the triangles tend to become mere point
triangles.
123. Fundamental spherical triangles when there is no
natural boundary. We now consider the first case when
1 1 1
-+ + > 1
p q r
and the orthogonal circle is imaginary.
If we stereographically project the w plane on to a sphere
which touches the w plane at the real centre of the orthogonal
circle, the fundamental curvilinear triangle becomes a spherical
triangle which we shall now denote by ABC.
The only possible solutions of the inequality are
(1) p = 2, q = 2, r = m; (2) p = 2, q = 3, r = 3 ;
(3) p = 2, q = 3, r = 4 ; (4) p = 2, q = 3, r = 5 ;
or equivalent results obtained by permutation of the integers.
We lose nothing by taking A, B, to be the correspondents
to the singular points a, c, b in the x plane.
We may thus have *f or the fundamental spherical triangle
any of the four figures which follow.
FUNDAMENTAL SPHERICAL TRIANGLES
167
The operation of inversion is now replaced by the simple
operation of taking the reflexion of each vertex with respect
to the opposite side. We see at once that the whole surface
of the w sphere is covered by the triangles and their images.
In the first case we have 2m triangles in the upper part of
the hemisphere and 2 m triangles in the lower part.
In case two we have a triangle whose area is ^ that of the
sphere, and by taking the six triangles with a common vertex
at A we have an equilateral triangle whose area is that of
the sphere : that is, we have the face of a regular tetrahedron.
In case three, which is just that of the triangle formed by
bisecting the angle G in case two, we have a triangle whose area
that of the sphere. By taking the eight triangles with
s
a common vertex at A we have the equilateral quadrilateral
whose area is ^ that of the sphere, that is, the face of a regular
cube. Its angles are each ,- ; and it is also the figure
o
168 PLATEAU AND CONFORMAL REPRESENTATION
formed by planes, through the centre of the sphere circum-
scribing a regular tetrahedron, perpendicular to two pairs of
opposite edges.
In case four we have a triangle whose area is T ^ that of
the sphere. By taking the six triangles, with a common
vertex at B, we obtain an equilateral triangle, whose area is
,jV that of the sphere : that is, a face of the regular icosahedron.
124. Summary of conclusions. When A, /z, and v arc
then the reciprocals of integers, we have found functions w
of the complex variable, which will conformally transform
the upper and lower halves of the x plane into the area
within the curvilinear triangles in the \o plane. To each
point in the x plane there will correspond, in the w plane, one
point in each triangle or in the triangle adjacent which is
its inverse. The real axis will be transformed into the
circular boundaries of these triangles.
Two different points in the x plane cannot h&ve the same w
to correspond to them. For by taking A, /x, and v to be the
reciprocals of integers we have provided against any over-
lapping in the w plane.
It follows that # is a uniform function of iv.
In the case where A -f /z -f- v > 1 there are only a finite
number of values of w which will make x zero or infinite ;
and therefore x will be a rational function of w. We could
express each value of w which makes x zero in terms of any
one, and thus obtain the numerator of the rational function.
Similarly we could find the denominator. As we only wish
to give a general explanation we do not enter into any details.
We have now shown how to represent the 10 plane, or its
equivalent sphere, on the x plane.
125. Representation of the x-plane on a given polygon.
To complete the problem of conformal representation in so
far as it bears on the problem of Plateau, we have now only
to show how the x plane can be conformally represented on
a given polygon. The' procedure is much the same as in the
problem we have just discussed, tout much simpler.
REPRESENTATION OF OHPLANE ON A GIVEN POLYGON 169
Let a, o, b be defined as earlier and let a, f}, y be three
real constants which are positive, and such that
a + /J + y = l. (125.1)
Let X = I** (x-a)"- 1 (x-l)f- 1 (x-c)^ 1 dx, (125 . 2)
J-oo
and let A be the position which X attains as x moving along
its real axis approaches a.
As x moves along the real axis in its plane from oo to a,
the argument of X is zero, so that it too moves along the
real axis of its plane. As x moves along the small semi-
circle with centre at a, the argument of X diminishes by OLTT.
As x then moves along the real axis to c, X moves along
a straight line AC to (7, the point which corresponds to c.
When x describes the semicircle at c, the argument of X
again diminishes by yir. Then as x moves along the real
axis from c to 6, X describes a straight line CB to B the
point which corresponds to b. X is now again on its real
axis ; and as x describes the semicircle at b the argument of
X diminishes by /JTT. Finally as x moves along the real
axis to +00 and then from GO to a, X describes the straight
line BA.
We thus have the figure
in the plane of X, and the upper half of the x plane is con-
formally represented by the area within this triangle.
By a transformation of the form X' = pX 4- q where p and
q are constants the triangle may be transformed into any
similar and similarly placed triangle in the plane of X ;
and thus the upper half of the x plane may be conformally
represented by the area within the triangle ABO which lies
in the piano pf X anywhere. *
170 PLATEAU AND CONFORMAL REPRESENTATION
We thus see, as before, that the pland'of x can be repre-
sented by a series of triangles in the plane of A", which will
cover it completely. But if there is to be no overlapping we
must have a, /?, and y to be the reciprocals of integers.
These integers must satisfy the equation
1
-
q
1
-
r
(125.3)
and we see that the only solutions of this equation are
p = 6, 3=3, r = 2 ;
p = 4, 3 = 4, r = 2 ;
j> = 3, 3=3, r=3. (125.4)
Wo thus have three cases
and we see into what kind of triangles the given polygon
must be decomposable in order that x may be a uniform
function of X.
We see that X is a doubly periodic function of x ; and
from the above triangles, and their images in the sides, with
respect to the opposite vertex, we can construct the period
parallelograms.
126. We have found corresponding to each value of ttf,
the complex variable of the sphere, a definite value of x.
This value of x will under certain circumstances which we
have considered be a rational function of w. To this value
of x we must choose/as its correspondent X, that value, or
those values, which lie within the given polygon. Since
REPRESENTATION ON A POLYGON 171
the values of w which lie on th# boundary of the spherical
polygon are to correspond to values of X lying on the
boundary of the plane polygon, and since these values of w
correspond to points on the real 'ftxis of x, we see that the
polygon must have its boundary made up of sides of the
elementary triangles in the X plane.
The principal results in the theory I have tried to explain
in outline are due to Riemann, to Weierstrass, and to Schwartz;
and my presentation is based on the treatises of Darboux and
Bianchi. The connexion of this branch of Geometry with
the Theory of Functions is interesting.
CHAPTER X
OKTHOGONAL SURFACES
127. A certain partial differential equation of the third
order. We now want to consider the theory of a triply
infinite system of mutually orthogonal surfaces ; and \ve
begin by considering the partial differential equation of the
third order
p -f q Y- + sech 2 x = q tanh x, (127.1)
where 20 = tan- _ (127 . 2)
v '
and z is the dependent variable and x, y, and w the inde-
pendent variables. [Here p, q, r, s, t denote respectively
We shall see that it is on this equation that we depend
when we wish to obtain the general system of orthogonal
surfaces.
Let ~ be any function which satisfies this equation, and let
-,
cx cy d
W ~ p cosh 2 x - h Q cosh 2 x - h -
1 dx * <)y ^w
then it is not difficult to verify that
VW- WV = F(pcosh 2 o;) . V. (127 . 3)
It follows that a function u exists which is annihilated by
the operators F and' W 9 and also a function v annihilated
by U and W.
A PARTIAL DIFFERENTIAL EQUATION OF ORDER 3 173
Wo may therefore regard x and y as functions of u, v, and
w, and we have
<$u <>u <H<> <>u <*u
, a* + -, y. 2 = 0, r #0 + r y ., + r = 0,
2 3 t/0
where the suffixes 1, 2, 3 respectively denote differentiation
with respect to u, v, w.
But from the definition of 11 and v we have
^^6 J . c)?C _ c^U . c)^6 <)iC
r + tan 6 . = 0, p cosh 2 a? r + q cosh 2 tc - + -- = 0,
^)ic ^y ^x <>y div
^v , A<*V ^ o ^v . ^v ^v
. - cot ^ x - = 0, w cosh 2 x - + a cosh 2 x ^ + - = ;
ex dy L dx J dy %w
and therefore it follows that
= iV 2 ~" 2/2 co ^ ^ = 0,
= 0, 2/3 <1 cosh 2 ^ = 0. (127 . 4)
We now see that
so that the equation with which we began becomes
3 = ^ sinh a; cosh <r, (127 . 5)
that is, 3 = 7/ 3 tanh x.
We thus have the three equations
#! + j^ tan = 0, .T 2 2/2 cot = 0, 3 = ?y 3 tanh oj. (127.6)
Now let
sin*
, rj = e~~
., - y-6' fc ~ " 2 '
cos ------ cos --r
2 A
so that tan 6 = ^ ,
then we can verify that
** * * ' (127.7)
171 ORTHOGONAL SURFACES
128. A solution led to when functions satisfying a set
of three equations are known. By retracing our steps wo
may verify that if we have any three functions which satisfy
these equations we shall be led back to the solution of the
equation of the third order. For we have clearly
x l + y l tan = 0, a* 2 ?/ 2 cot = 0, 3 = y 3 tanh x ;
and therefore = a 1 , 17, c- = # V.
^u L <)v
N OW A = S i n 2 Q u+ cos 2 6V, ~ = sin 6 cos 6 (V- U) ;
and therefore we can verify that
A ^ = _^*_ ,
<>2/ cosh 2 ^ <)aj cosh 2 x
so that # 3 = > cosh 2 #, 2/a = 5 cosh 2 .r. (128.1)
From # 13 -f 2/13 tan 6 + y l sec 2 0^ =
we verify that
U (p cosh 2 s) + tan CT (5 cosh 2 a)
= (cot -f- tan $) q sinh x f cosh r/:,
and therefore tan 2 5 = 28 + 2 7 tanha! . (128.2)
r ^ + 2 ^> tanh a;
Finally we see that --- = TF,
oiy
and thus the equation 3 = y 3 tanh x
or 3 = ^ sinh a? cosh a/
becomes Wd = g sinh a? cosh x. (128.3)
The equations
are thus connected in the way we have described with the
partial differential equation of the third order.
129. The vector qaq~ l , where a is a vector and q
a quaternion. We now pass on to the geometry which we
associate with these three equations.
If we are given any vector an& any scalar quantity we can
A ROTATED VECTOR 175
take the vector to bo r sin . e and the scalar quantity to be
r cos 0, where is a unit vector.
Let q = r (cos + e sin 0), (129.1)
then q is called a quaternion, e is called the axis of the
quaternion, 6 is called its argument, and r is called its modulus.
A quaternion is thus just the ordinary complex variable of
the plane perpendicular to the axis of the quaternion.
We have q~ l = r~ l (cos 0-esin d). (129.2)
Any other vector may be written
xt' + ye, (129.3)
where x and y are scalars and e' is some unit vector at right
angles to e.
We see that q(xt' + ye)q~ l
is equal to x(e' cos 20-f e" sin 26) + yt, (129. 4)
where e" is the unit vector perpendicular to 6 and e'. That
is, if a is any vector, q(X q-i ( 12 g . 5)
is just the vector oc rotated about the axis of the quaternion
through an angle double the argument of the quaternion.
130. Passage from set to set of three orthogonal vectors.
Let us now consider the quaternion
7= l+i + i)j + (k, (130.1)
where i, j, k are fixed unit vectors at right angles to one
another and , 17, f are any three scalar functions of the
parameters u, v, and w.
Dq~ l = l-i-r,j-(k,
where D = 1 + 2 + rf + * . . (130.2)
Let X = qiq~ l , /JL = qjq~ l , v = qkq~ l , (130 . 3)
then A, //, v will also be three unit vectors mutually at right
angles to one another, no longer fixed vectors but depending
on the parameters u, v 9 w.
Any system of mutually orthogonal unit vectors can be so
defined. *
176 ORTHOGONAL SURFACES
We easily see that
(130.4)
Now , WVir^tfiT 1 ,
and therefore W 1 !?!?" 1 = 9V/ "^
It follows that
* + Oh - tf i
(130.5)
From qi ~ \q
we have 7V/~ 1 ^""^ r /i7~ 1 := ^11
and therefore f/i'/" 1 ^ ^ r /V/~ 1 = ^r
It follows that, since
/zi> VJJL = 2A, z/X At/ = 2/z, A/z /x\ = 2y,
Z)A 1 =-2( 7l -^ 1 + ^ 1 ) J , + 2(^-^ 1 +-^ 1 )^ (130.6)
Let /S-
4 Zty' " = | 3 - itf, + ^ a) 4 D?'" = 7,3 -
then we have proved (130.6) that X, = pr' vq'; and similarly
we prove the other equations of the system
(130.7)
It may be noticed that the q', q", q'" as here defined have
no direct relationship to the quaternion q where
q=
If >' =
SETS OF ORTHOGONAL VECTORS 177
we see that A 1 = co / X, ^ = o>'/*> i/ 1 = a/j/,
A 2 = oPA, // 2 = o>'>, i/ 2 = o)" v,
A 3 = o/"A, ^ 3 ==o/ 7 >, j/ 3 = G>"V (130.8)
and we can easily verify from the formulae given that
a>' = 2, / 7?- 1 , a>" = 2<M- 1 , '" = 2 (M" 1 (130.9)
where g is the quaternion.
We also have as we proved earlier [see 90] the equations
o)'" 2 a>" ;J o0"o>'", 0/30)'"! = &"'<*>'> 3>"i fi/2 == ^'^
(130.10)
The angular displacements of the vectors >, /*, *> regarded
as a rigid body are ^^ ,"<&,, n'" c l w . (130 . 11)
131. Rotation functions. So far we have been consider-
ing a system of three unit orthogonal vectors of the most
general kind depending on three parameters, and we have
seen how they depend on the quaternion
l+i--iM + fc.
We now want to consider the particular system characterized
by the property that p' q" = T f " = Q, (131.1)
that is, by the property that , 77, satisfy the three equations
(127 . 7) which in 128 we connected with the partial differential
equation of the third order with which we began our discussion.
We now have from (130 . 7)
\ l = /ir'-rj', A 2 = fir", A 3 = -vq"',
K = -Xr', fr = n >"-\r", to = n>'">
v, = X 9 ', ^ 2 = -w", v. = A^'- W /".
It will be convenient to write
when the above equations become
= 0, /* 2 + i/(32)+X(12) = 0,
(131.2)
178 ORTHOGONAL SURFACES
The six functions
(23), (32),- (31), (13), (12), (21)
we shall call rotation functions. They are connected by the
laws
(21) a +(3l)(32) = 0,
(23) 1 =(21>(13), (31) 2 = (32) (21), (12) 3 = (13) (32),
(32), = (31) (12), (13) 2 =(12)(23), (21). = (23) (31),
(131.3)
as we can at once verify from the equations satisfied by
A, fi y v. We can express these rotation functions, as we have
done, in terms of , 77, and their derivatives.
132. A vector which traces out a triply orthogonal
system. Now consider the system of equations
(132.1)
where a, /?, y are scalars to be determined by these equations.
We see at once from the set of conditions
(23), =(21) (13), (31) a =(32)(21), (12) 3 = (13) (32),
(32), = (31) (12), (13) 2 = (12) (23), (21). = (23) (31)
that they are consistent.
Let a, ft y be any three functions which satisfy them, and let
2 = aA + jfy + yy. (132. 2)
We have z l = ( 1 + j8(21)-f y(31))A,
(132.3)
and therefore the vector z traces out a triply orthogonal
system of surfaces.
Conversely we see that there is no triply orthogonal system
of surfaces which cannot be obtained by this method.
LINES AND MEASURES OF CURVATURE 179
133. Lines and measures of curvature. If we take
), 6 =0 2 + y (32) +
wo have z l = a\, z. 2 = bp, z. A = cv,
and we see that
, = 6(21), 6 3 = c(32), c 1 =
3), (133.1)
and that these three last equations together with
a = -3u, 6= C<2 (133.2)
(13) ' (23) * '
are equivalent with the first six.
The three orthogonal surfaces are
u = constant, v = constant, w = constant ;
the unit vectors parallel to the normals at the extremity of
the vector z are respectively A, ^, v.
We have ^ = ^> ^ = ( 133 - 3 )
and therefore the curves along which only v and iu respectively
vary are the lines of curvature on the surface u = constant,
and its principal radii of curvature are
(133<4)
We thus have the fundamental theorem about lines of
curvature of orthogonal surfaces, viz. that they are the lines
in which the two other surfaces intersect one of the surfaces.
If we consider the curve in space % along which only u
varies, and if we suppose its principal normal to make an
angle 6' with the vector /*, *and p' and cr' to be its two
180
ORTHOGONAL SURFACES
curvatures, we have in the notation we used in considering
curves in space (see 94)
cos0' sin 6'
(133.5)
Now aX = X x , a/i = // 1 , av = j^,
since acZu is the element of arc of the curve, and as we have
we must have
, (31) =
"+ > = 0,
(T
Thus considering the three curves we get
6'+ i = 0,
"
(23, _
^ ' ""
(32) =
"' + r,= 0,
or
(133.6)
and we thus see another interpretation of the rotation functions.
In the figure here given A, B, C represent the points where
e"
" A
the vectors X, /*, ^ intersect the unit sphere whose centre is
the origin ; that is, the points* where parallels to the three
MEASURES OF CURVATURE 181
tangents to the curves intersect the sphere ; and A', E\ C' the
points where the parallels to the corresponding principal
normals intersect the sphere.
The principal radii of curvature of the surface u = constant
were, we saw, /> c
(72) and (is)'
that is,
-p'"
and therefore the measure of curvature is
, / ' /
7)
///// - - //"///
P P P P
Similarly we have
- CO * C ' A ' '" - CosA ' B '
K" -
~
p'"p'
Again, from the formulae
6'+- f = 0, d"+4 = 0, d'"+-*,, = (133.9)
0" CT CT
we at once see that, if a line of curvature is a plane curve, its
plane cuts the surface at the same angle all along it.
134. Linear equations on whose solution depends that
of the equation of the third order. We now return to the
equation of the third order (127 . 1),
12 * d 12 ^ ^
p coslr x + q cosh 2 x + ~ = q smh x cosh a?,
dsc 2 ty 1*10 * '
where 2 5 = tan^ 2g+2 ^ tanh .
r t + 2p tanh a;
Suppose that z is any integral of this equation : we may
suppose it expressed in the series
* =/( 20 + ^0(0, 2/)+^ 2 ^(, 2/)4- ..., (134 . 1)
and if the integral is a general one we may take / to be any
arbitrary function of x and y.
We shall show how wh8n / is a known function the
182 ORTHOGONAL SURFACES
function <f> depends for its determination on a differential
equation of the second order.
Let
and let /2 = cosh 2 x ~- ~ 4- cosh 2 # ~
ox ex ^y cy
The equation which determines is then
Qf2P-Pfi,Q-2 sinhacosh x (P 2 + Q 2 )
(134.2)
Now let tv = W' + ^'Q, where w is a small constant whose
square may be neglected, then
s = /+W' + w'(0 + 2' ^r) + ..., (134.3)
and by solving a similar equation to the above with /-M' f () </>
substituted for /we should find
+ 2?^, (134.4)
and thus obtain ty.
Proceeding thus we see the system of linear partial differential
equations on whose solution we depend for obtaining the
coefficients of the different powers of w in the series for z.
A particular solution of the equation of the third order
would bo obtained by taking/ to satisfy the equation
QflP-PflQ = 2 ^ sinh x cosh x (P* + Q*)> (1 34 . 5)
ij
when we could take to be equal to /.
135. Synopsis of the general argument. It may be useful
at this stage to give a re8um6 of the general argument.
z is a function of x, y, and w which satisfies the equation
<>0 ^0 10 ^6
p- h q rr- V secrr# - = q tanh x,
1 *
SYNOPSIS OF THE GENERAL ARGUMENT 183
, o/j x i
where 2 6 = tan"" 1
A
r t + 2 p tanh x
W = 2> cosh- x ^~- + q cosh 2 a? + ^- ;
* d# 2 c)7/ aw
and ?6 and v are defined by
Vn = 0, T^ = ; L"y = 0, Wv = 0.
We can now express a, ?/, and in terms of if, v, and ?; ;
and having done so we define , 77, by
y-f<9 . + 6
cos - sin
~
__ _ _
7/-fl' - ?/-g' - 2 '
cos cos ^
^j 2i
and we have
The functions ^, 77, now define a quaternion
'/= 1+^-f W + ^%
where /, j, k are any fixed unit vectors at right angles to one
another.
Three unit vectors mutually at right angles are now de-
fined by x = <]iq~ l , /* = </^/" 1 , v qhf- 1 ,
where D([~ l = lirij k
and 7)= i+* +1? 2 + ^
These vectors are not fixed.
We have
(31) = 0, //> + */ (32) + A (12) = 5
and thus the six rotation functions
(23), (32), (31)* (13),' (12), (21)
184 ORTHOGONAL SURFACES
are defined. These functions satisfy the conditions
(31) 8 +(I3) 1 + (23)(21) = 0,
(31)(32) = 0,
^ = (21) (13), (31), = (32) (21), (12) 3 = (13) (32),
(32), = (31) (12), (13) 2 = (12) (23), (21) 3 = (23) (31).
The vectors X, /*, v are parallel to the normals at the ex-
tremity of some vector z depending on three parameters
which traces out the three orthogonal surfaces
n = constant, v = constant, w constant.
This vector z is defined by
z = X + /3/i + yv,
where a, j8, y are scalars to be determined by the six equations
y 2 = j8-(32).
Corresponding to each solution of this equation system we
obtain a system of orthogonal surfaces, and the different
systems thus obtained have the property of having their
normals parallel at corresponding points.
If a = ai + (21) + y (.31), b = /?, + y (32) 4- a (12),
c = y a + a(13)+/8(23), (135.1)
then c 1 = aX, z% = bfi, z.^ cv,
and a, = 6(21), 6 3 = c (32), c 1 = a(13),
3 = r(31), 6 1 = a(12), C 2 --= fc (23),
so that the ground form for the Euclidean space is
ds 2 = a 2 rfu 2 -I- 6 2 cZi; 2 + c*dw\ (135.2)
136. An alternative method indicated. The functions
c(, 6, c of ^, v, w must satisfy certain conditions which can
at once be obtained by expressing the rotation functions in
terms of a, b, c and their derivatives and using the conditions
which the rotation functions "must satisfy. But we can
AN ALTERNATIVE METHOD INDICATED 185
more rapidly obtain these conditions by just saying that the
space defined by d 8 z - a *d u * + I>*dv 2 + c*dw* (136.1)
is flat, and therefore (rkih) = 0. (130 . 2)
The conditions then arc seen to be
, 7 PI 7 <>>
"
These six equations if we could solve them would equally
lead to orthogonal surfaces, and this is the usual method by
which the problem of orthogonal surfaces is approached. There
seems, however, to be an advantage in making the whole
theory depend on one equation of the third order as we
have done.
137. Three additional conditions which may be satisfied.
We now wish to consider a special class of orthogonal surfaces,
and we begin by inquiring whether there are any rotation
functions which, in addition to satisfying the nine necessary
conditions which all rotation functions must satisfy, also
satisfy the three additional conditions
(23) 3 +(32) 3 + (21)(31) = 0, (31), + (13) 3 +(32)(12) = 0,
(12). 2 + (21) 1 + (13)(23) = 0. (137.1)
If we take
(23)EEu; + (31) = ^
(32) =x-, (13)E
and 2 u' = v + iv, 2 u' = w -f u, 2 iv f = n + v,
and, /being a function of the parameters t</, v', ty', denote
respectively by / x , / 2 ,
186 ORTHOGONAL SURFACES
we see that the twelve conditions which the rotation functions
now have to satisfy are expressed by
2/4 = -
& = -27;s, 7, 3 = -2
*3 = -2^ f 17! = - 2^, ^ = -2iya?. (137.2)
Now it is easily seen that these equations are satisfied by
taking
(23) = 4V%, + av%. (32) =
(12) = -|/K; + ^V 3p (21) = i
(137.3)
where F is a function satisfying the 'complete' system of
equations
K ;14 + 2 v/CK 4 V 42 = 0, V. M + 2 VV~V~V K = 0,
;K 4 = o, ^+2^1^7 = o.
(137.4)
\Ve thus see that such rotation functions exist. A particular
solution of such a system of equations would be obtained by
toeing F< = 0) ^ a+2 v%lK ll -=0; (137.5)
and in this case
(23) = (32), (31) = (13), (12) = (21). (137.6)
It may be shown that this solution corresponds to the
particular solution of the original equation of the third order
when we take z to be independent of iv.
138. Orthogonal systems from which others follow by
direct operations. We must now consider the special property
which the orthogonal surfaces will have which correspond to
rotation functions satisfying the twelve conditions. We return
to the original variables n, v, w in what follows.
Let a, /3, y be any scalars which satisfy the equations
0,-=y (13), ia =a (21), y 2 = j8(32), (138.1)
SPECIAL CLASSES OF ORTHOGONAL SURFACES 187
and let a, b, c be defined by
o = a 1 +/8(21)+y(31), b = a + y (32) -f a (12),
c = y a + a(13)+j8(23).
We then have
(138.2)
Let a' = </! + 6(12) + c(13), 0' = 6 2 + c(23) + rt (21),
/ = c 3 + a(31)+&(32).
We can at once verify that
^3=7' (13), ^ = ^(21), y' 2 = /8'(32); (138.3)
and therefore c' = a'X + /S'/i-f y V
will trace out another system of orthogonal surfaces. This
second system is thus obtained from the first by direct
operations not involving integration. We thus see that when
we are given any one system of orthogonal surfaces of this
particular class we can deduce by direct operations an infinite
system of such surfaces.
CHAPTER XI
DIFFERENTIAL GEOMETRY IN ?i-WAY SPACE
139. Geodesies in w-way space. In order to see what
kind of geometry we may associate with the ground form
of an 7i-way space, we naturally think of the simple case
when n was 2, and the space a Euclidean plane. The most
elementary part of that geometry was that associated with
straight lines ; that is, the shortest distances between two
points. We are thus led to consider the theory of geodesies
in our 7i-way space.
We have
. , f
2 T- = a ik -7 +a ik ~i --, j -r
da ll * ds ds ds da ds ds
d / dx- dx 1f . \ d
= ds ( rt ds SX >< + a * -* **') - 8X >< ds
For a path of critical length therefore we must have
. ,
ds \ it d8 ds < d~ 8 ~ T^ ds 'ds
Now ( 6)
d Za ik dx p dx p// .
and therefore
dx: daw t
GEODESICS IN M-WAY SPACE 189
It follows that
7? =
and therefore a it ^ + (ikt) d p ^ = 0. (139.2)
' ds* as as
Multiplying by a l P and summing we have
X f + {ikp} _ -^ = 0. (139 . 3)
We thus have TI equations wherewith to obtain the coordinates
of any point on a geodesic in terms of the length 8.
But the equations are differential equations of the second
order ; and in general we can only solve them so as to obtain
the coordinates in tbe form of infinite series. This is a
practical difficulty and one of the reasons why we cannot
have the same kind of knowledge of the theory of geodesies
in Ti-way space that we have in Euclidean geometry of
straight lines.
The direction cosines of an element of length in ?i-way
space are defined by
dv
ti> - ax p n _ i n / 139 4 \
i* ; * iJ 1 ,,/(/ i A o jj TI i
5 ds ' L v ;
Going along a geodesic, therefore, we have
and we see that, unlike the direction cosines of a straight
line in a plane, associated with the form
ds* = dx\ +dxl,
these direction cosines vary as we pass along the geodesic.
Thus we are familiar with the difficulty of keeping to the
shortest course between two given points at sea, viz. a great
circle. In this case the differential equations are soluble in
finite terms; but even with* this advantage we should need
190 DIFFERENTIAL GEOMETRY IN M-WAY SPACE
a continuously calculating machine to find the direction
cosines at each point of the course. If the ocean instead of
being spherical were ellipsoidal, we should not even have tho
advantage of being given the equations of the geodesic in
finite form, and the difficulty of keeping to the shortest course
would be even greater.
Now if we had built up our plane geometry by using the
form d8* = dxl+x*dxl,
the direction cosines of a straight line would also have varied
from point to point of the straight line and yet we would not
say that the direction of the straight line varied from point
to point. The navigator on the ellipsoidal ocean might hope
till he had learnt a little more geometry to mend the want
of constancy in his direction cosines as the plane geometer
could mend his by a proper choice of coordinates.
He could not mend this want of constancy by any choice
of coordinates, but though the direction cosines change in
passing along a geodesic there is no need to think of the
' direction ' as changing.
We will then say that the direction in an 7i-way space is
the same all along a geodesic.
140. Geodesic polar coordinates and Euclidean coordi-
nates at a point. We recall the fact ( 2) that any w-way
space may be regarded as lying in a Euclidean ?*-fokl where
r = %n(n+ 1), and that the vector z which lies in this r-fold,
depending on the n parameters x l ...jc ny has the property
that its extremity traces out our ?i-way space.
In the 7i- way space, unless it happens to be merely a
Euclidean space, we cannot think of a vector as lying in it :
it is only tho extremity of the vector with which we are
concerned.
But at any particular point of the 9i-way space there is
a Euclidean w-fold which we may usefully associate with
the point.
Let z be the vector to the point under consideration, and
let z lt .*z n be its derivatives at the point with respect to
x l ... x n , the parameters of the jfoint.
TANGENTIAL EUCLIDEAN W-FOLD TO AN W-FOLD 191
Let 1 ... w be the direction cosines of any element of the
n- way space at the point so that
!** = 1, 040.1)
then the vector defined by
= 1 s l + . ..+%, (140,2)
will lie in the Euclidean Ti-fold at the point. It will clearly
be a unit vector sinco
We shall call this Euclidean '}i-fold in which lies the
tangential 7i-fold at the point.
The coordinates of any point in the tangential ?i-fold ma}''
be taken as t . . . g n , where
& = B ; , (HO. 4)
s being a scalar.
We establish a correspondence between the points of our
?i-way space and the points of the tangential w-fold by taking
the coordinates of the 9* -way space to be a ... M .
Consider the geodesic which starting at the point under
consideration has the direction cosines l ... n .
From the equation of a geodesic
ff . --- .
ds d*
we see that the current coordinates are given by
' > (140 . 5)
>
where s is the arc from the initial point.
Let z 9 be the vector which traces out the ?i-way space at
the point x' l ...x' n> and let 9 denote the same vector expressed
in terras of the coordinates & ... n -
We have [see 4 for the notation]
n = ^(*;-{^}r+'-o-
It follows that
192 blFFERENTIAL GEOMETRY IN n-WAY SPACE
and therefore the transformation formula is
a>'n = <* ik -(kpi) l '8-(i\k)?8 + ..., (140.6)
where + ... refers to higher powers of s.
We thus have at the oriin
that is, in the coordinates we have chosen the first derivative
of each of the coefficients in the ground form vanishes.
It follows that in this system of coordinates, which
establishes a correspondence between the points of the
tangential n-fold and the points of the ?i-way space, every
three-index symbol of Christoffel vanishes at the point under
consideration.
As regards the four-index symbol (rkih)' we have
= (rkih). (140.8)
We may call this transformation a transformation to geodesic
polar coordinates at a specified point.
We can combine the transformation with any linear trans-
formation in the tangential 71-fold. To do this suppose
x 1 ... x n to be the original coordinates, taken to be zero at the
point to be considered.
Let i = ^vX* (HO. 9)
where Cy.... denote constants.
We can now so choose these coordinates as to make the
coefficients take any assigned values at the point We can
then apply the geodesic transformation, and can thus arrange
that the coefficients a ik may have any values we like (pro-
vided the determinant is not zero), and at the same time
have all the three-index symbols vanishing at the point.
In particular we caiiuso choose the constants that
'<&='< (140.10)
EUCLIDEAN OR GALILEAN COORDINATES AT A POINT 193
at the point and that the three-index symbols may vanish.
Such a system of coordinates may be said to be the Euclidean
coordinates of the n-way space at the point.*
141. Riemann's measure of curvature of 7i-way space.
If we take the transformation
where , 77, ... are fixed vectors at the point, we find that
a' 22 = 'S*, (141.2)
and
(141.3)
and therefore
(1212)'
(141.4)
Now let us consider the expression on the left-hand side of
this Aquation.
In general the four-index symbol as applied to 7i-way space
is not the same thing as when applied to the lower space in
which the coordinates whose integers do not occur in the
symbol are put equal to constants. But in geodesic polar
coordinates at the point the equality holds, since ,the three-
index symbols vanish.
It follows (see 24, 37) that^ the expression on the left is
the measure of curvature at the point of the two-way surface
formed by keeping all the geodesic coordinates constant except
two. The expression on the right is therefore the measure of
* This system of coordinates has jjeen calld the system of Galilean cq-
ordinates at the point.
2843 C
191 DIFFERENTIAL GEOMETRY IN n-WAY SPACE
curvature of the geodesic surface that is, the surface formed
by the geodesies through the assigned point, which touches
at the point the Euclidean plane generated by the two vectors
and rj.
This is Riemann's measure of curvature of the 'ft- way space.
We see how it is connected with Gauss's measure of curvature,
and we should notice how in this respect the tangential
7i-fold takes the place of the mere tangent plane when n = 2.
In the flat Ti-fold we consider all the Euclidean planes by
taking 1 any two vectors in the Ti-fold. We see that these
two-way surfaces have different curvatures and so different
geometries.
142. Further study of curvature. The Gaussian measures
for geodesic surfaces. Orientation. We have now obtained
Riemann's measure of curvature and have seen how it is con-
nected with Gauss's measure of curvature of a surface.
We must now consider this curvature from another point
of view.
We saw that we were to consider the direction to bo the
same at all points of a geodesic in u-way space. This leads
us to define two 'parallel' displacements at neighbouring
points x l .. f x n and x 1 + dx 1 ... # + dx n as displacements whose
direction cosines 1 ... n and *-f d l ... n + d n are con-
nected by the equations
dP+ {ikp} gdx k = 0. (142 . 1)
Thus in this sense of 'parallel' the tangents are parallel at
all points on the same geodesic.
It may be noted that the equation defining parallel dis-
placements does not entitle us to say that
If this equation system held, the tensor component P would
be annihilated by every operator 1,2, ... TI, and therefore
which could only*be true in flal space.
FURTHER STUDY OF CURVATURE 195
Let be any vector in the tangential ?t-fold at x^ ... oc n and
+ d be tho 'parallel' vector in the tangential ?i-fold at
n\ -f dx l . . . x n -
We have df = {'(*.<* + [i&
As we pass from the point of departure with an assigned
value for , and the vector is carried parallel to itself, its
value at any other point is defined by the integral
3. ik Jx k , (142.2)
and this value depends on the path of integration.
Consider the small parallelogram in the ?i-way space whose
edges are parallel to the vectors and 77, the lengths of the
edges being respectively a and 6. Wo want to find the change
in f by integrating round the parallelogram.
We have i = = '-
where SjCp is the increment in the coordinate, neglecting
powers of small quantities of the second order, and
Z-tfc = (3-ifc)o + 0-tt/*+ (*^/ 3/fc+ (Ml S-ttL&V
and therefore
We thus have
*'* = ^'*-)o+ [r (*-^+ {i>j -
On the first edge at a point distant s this is
K f *-tt)o+[^ (*+ {*/*'} -.)n
on the second edge it becomes
tf f s-)o+[^ (-
+[r(^^
The change in ^ bj' integrating
196 DIFFERENTIAL GEOMETRY IN tt-WAY SPACE
along the two edges is
) W]o-2 ( 142 3 )
If we had integrated in the opposite sense along the other
two edges we should have interchanged and 77, a and 6, and
we thus see that the change in by going round the parallelo-
gram in the same sense is
K'(-tt M -*-< M *)*V]o&>
that is, [f f {#Ms,V] &.
The change in ^ is therefore
If is the angle between and
(142.5)
Let us now consider how the angle d is changed, if, keeping
77 fixed, we carry parallel to itself round the parallelogram.
ab.
It follows that <5d divided by the area of the small
parallelogram is equal to
~ 6)
That is, 5^ 'divided by the area of the small parallelogram is
equal to the curvature of the geodesic surface which touches
the Euclidean plane generated by the two vectors formed by
the sides of the parallelogram.
From the equation
tff=f'*-ttk*
we see that the rate of Change of f in parallel displacement
FURTHER STUDY OF CURVATURE 197
in the direction of the vector is ^C Zt il^ We ma J
express this result in the notation
Thus we have . = fe *'
It follows that
and therefore
(142.7)
Kieumnn's measure of curvature may therefore be written
^-fcf*
.^.. ____. (142.8)
||25""l?2l
Again the rate of change of in displacement along the
vector is just f 1 ^ 1 + ... + n <z> and therefore the vector
itself may be written 0.. Here we may notice that the
vector z unlike the displacement vector is not a vector in
the 7i-way space but only in the containing ?*-way flat space ;
the vector on the other hand marks a direction, or displace-
ment, in the n-way space, although it has only an elemental
length in this space.
There is then as regards the vector z in its parallel displace-
ment just the ordinary Euclidean ide'a of translation.
198 DIFFERENTIAL GEOMETRY IN ,71-WAY SPACE
Riemann's measure of curvature may therefore be written
-^- ^ (142.9)
This is Gauss's measure of curvature -of the geodesic surface,
made up of the singly infinite system of geodesic curves
drawn through an assigned point at which we require the
measure of curvature: the curves at the point all touch
the Euclidean plane generated by the vectors ^ and 77.
Riemann's measure of curvature has an ' orientation ' given
by the vectors ^ and 77 ; and at the assigned point, by varying
this plane, we get the different Gaussian measures.
143. A notation for oriented area. So far we have in
using vectors only considered their products as scalar products.
There is another product which we ought now to consider.
When 7i = 3 and the ground form is that appertaining to
Euclidean space, we know what the vector product means and
how useful it proved in Geometry, but it does not seem to be
capable of useful extension. We shall now think of the
product of two vectors ^ and 77 as defining an area in the
Euclidean plane formed by ^ and 77. This area has then an
orientation, and we shall understand by 77 the area of the
parallelogram whose edges are the and 77 drawn through
the point.
The angle the vector makes with the vector 77 being 0, by
parallel displacement of the vector round the parallelogram
whose edges are in the directions and 77, and whose lengths
are ds and Ss, this angle is increased by
z tt~'w~~ z t'<i z i]t
which may be written
^ /* _ /y /?
z tt z w z ft"nt
dzSz, (143. 1)
'
z,z t z z
where dz, Sz represent* the sides ol the small parallelogram in
magnitude and direction*. *
ORIENTED AREA 199
We should notice that area has a sign as well as a magni-
tude : we express this by the equation
17 + 1^ = 0. (143.2)
144. A system of geodesies normal to one surface are
normal to a system of surfaces. If the direction cosines of
a geodesic are l ,... w > we have seen (139.5) that the
equations of a geodesic are
The geometrical interpretation of these equations is that
the tangent remains ' parallel ' to itself as we move along the
geodesic.
We can put the equations in another form,
and therefore
since -r
as
Now -(t
and tberefore the equations of a geodesic may be written
We now wish to consider the expression
T , = **
Wo know, from the theory of differential equations, that the
necessary and sufficient condition that T x dx^ may be rendered
a perfect differential by multiplying by a factor is that
; 2 *" T )
should vanish identically for* all values of A, /*, v ; and that
200 DIFFERENTIAL GEOMETRY IN n-WAY SPACE
the necessary and sufficient condition that T^dx^ may vanish,
wherever (f>(x 1 ... x n ) = 0, is that
2\
A
should vanish for all values of X, /z, v wherever
Geometrically interpreted these are the conditions that the
curves d ,j n
^==^==.,.= ^ (144.3)
should (1) be cut orthogonally by a system of surfaces, (2) be
cut orthogonally by the definite surface
(,... ) = <>. (H4.4)
We are now going to prove that if the curves are geodesies,
and if the condition (2) is satisfied, (1) is satisfied also.
\ rp *^rn
Let -.Jsta, (144.5)
and T p (97-) + T q (rp) + T T ( m ) = [p, q, r]. (144.6)
Since
A JL = -L. .* *? *
ds ^x~ ~~~ Zx ds dx <>jc'
cts v XQ (jts * u x^.. cts
Now
~
GEODESICS NORMAL TO SYSTEMS OF SURFACES 201
It follows that
Now
and therefore \ve see that the first term in this expression
vanishes, since A and /z are interchangeable.
We therefore have
It follows that
d
A. A.
[7- A. r]) + ||- ((rj>) r x + [r, A,
* >
D d
202 DIFFERENTIAL GEOMETRY IN tt-WAY SPACE
Since a \^i^ *>
we have *r A = 1,
and therefore T K ^- + x ~ - = 0.
It follows that
d r T
Suppose now that over a given surface
[p, q, r] = 0.
We then have
d r -,
fc[P>1.r]
= (qr) (p\) + (rp) (q X) + (pq) (r A).
Since [p, q, 7 1 ] is zero for all values of the integers over the
given surface, we have
=0,
0,
(sr) 9
(7*)>
0,
and therefore
(rs),
0,
M>
(F")
that is, (pq) (rs) + (^r) (/?) 4- (rp) (g) = 0. (144.10)
It follows that if [p, g, r] is zero over a surface it is zero
everywhere, arid therefore if a system of geodesies are normal
to any one surface they f are noimal to a system of surfaces.
(P9)
GEODESICS NORMAL TO SYSTEMS OF SURFACES 203
The direction cosines of the system of geodesies therefore
satisfy the equation system
It follows that K^aPi^^-aP^
= a qr gi r = 1, (144.11)
and therefore "nttf = P - ' ( 144 12 )
145. The determination of surfaces orthogonal to
geodesies and of geodesies orthogonal to surfaces. We can
now find the equation which (/> must satisfy when the surfaces
= constant
are those which cut the geodesies orthogonally.
We have ^> = u P f ~-^=- (145.1)
S V '
The equations of a geodesic being (131) . 5)
d
we must have
that is, aV --A~ a** -= = 0,
and therefore P f fc( ,a tx , A ) = 0,
or
=0. (145.2)
Expanding, A (0, <^) = ___ A(^, A
2 A ()
that is, -r
** q
204 DIFFERENTIAL GEOMETRY IN n-WAY SPACE
It follows that
and therefore A (0) must be a function of $.
Without loss of generality we may therefore say that
A(0) = l. (145.4)
We thus see that if we take any surface and consider the
geodesies drawn from every point on it perpendicular to
the surface, they are cut orthogonally by a system of surfaces
(f) = constant, where A (0) = 1.
In ordinary Euclidean space this is the theorem that the
lines normal to any surface are cut orthogonally by the
surfaces (f> = constant, where
Conversely, let be any integral of the equation A (</>) = 1 ;
then we shall show that the orthogonal trajectories of the
surfaces (f) ~ constant will be a system of geodesies.
It will be convenient now to think of an (>i-h l)-way space
and to take, instead of the variables x l ... tf n +i> a new system
of variables y l ... y n+l , where
y + iS0, (H5.5)
so that A(2/ n+1 ) = l,
and to choose y l ...y n as n independent integrals of the
equation A (y n+l , y) = 0. (145.6)
Let the ground form of the (n -f l)-way space be
1>K<lynlyit* 1 = 1, 2,... n+l. (145.7)
Since A (y ll + l ) = 1 and A (y n+l , y r ) = if r ^ (n+ 1),
we see that 6 n+1| n+1 = 1, & n+1| r = 0,
and therefore the ground form is
rfVi + 6 <A<fyfc> 1- = l> 2 > - ^ ( 145 8 )
It will therefore be convenient to take as the ground form
in the (TI-J- Ij-way'fcpa^e
du*+btydXidx kt i = 1 ... 7i, (145 . 9)
where 6^ is a function of x l ... cc w and u.
GEODESICS NORMAL TO SYSTEMS OF SURFACES 205
The surface ih = is any arbitrary surface in the (n + 1)-
way space, and when u = we may write b ik = a^ . We
may consider a^dx^lxj, (145.10)
to be the ground form of the u-way space deduced from the
(71 + l)-way space by putting u = 0.
The lower ground form may be said to be the ground form
of a surface in the higher space.
By varying u we obtain a series of surfaces cut orthogonally
by the curves whose direction cosines are given by
l = 0, 2 = 0,...^ = 0.
These curves are geodesies, since
{n+1, ti+1, p] = 0, p = 1 ...n.
It will be noticed that the first of the surfaces cut orthogonally
may be any whatever, but the other surfaces are given by
A(tt) = 1. (145. 11)
When we know the geodesies normal to u = 0, we know the
whole series of surfaces which are cut orthogonally, or at
least can find them by quadrature, since
'' = < (145.12)
We obtain the geodesies, on the other hand, by the solution
of the linear equation A (u, v) = 0, when we know u.*
146. A useful reference in (n + l)-way space. We have
shown (145.9) that the ground form of any (/i+l)-way
space may be taken to be
tin* + bftdxjdxjg , * = 1 . . . n,
where b^ is a function of x l ... x n and u.
The surface u = is any surface whatever in* the (u+1)-
way space. By drawing the geodesies perpendicular to this
surface we obtain a series of curves which are cut orthogonally
by the surfaces u = constant, u being the geodesic distance
of any point from the surface u = 0.
* [At this point hi the author's MS. Iliere ia a memorandum ' New
Chapter '.]
206 DIFFERENTIAL GEOMETRY IN 71-WAY SPACE
The surfaces u = constant are said to be parallel surfaces,
and we have A (u) = 1. Travelling along any of the geodesies
from the surface u = 0, only u varies.
It will, however, be found useful to consider a more general
system of surfaces in the (n+ l)-way space.
We therefore consider any system of surfaces whatever in
this space, u = constant, where we no longer have A (u) = 1,
and by taking the orthogonal trajectories of these surfaces,
as the parametric lines
x l = constant, ... x n = constant
we may take the ground form of the space to be
2 <Zi6 2 + b ik dx { dx k , i = 1 . . . n. (146.1)
The orthogonal trajectories are now no longer geodesies.
The function b ik depends on the coordinates x lt .. oc n arid u,
and, when u = 0, b ik = a^..
We now wish to consider the two round foi'ms
dxid^ (146.1)
and ^i k dx { d^ k (146.2)
in connexion with Christoffel's symbols, where after calculating
their values for the higher space we put u 0. We can
obtain the special case of parallel surfaces by putting 0=1.
We shall thus be shown how to generate the (n+ l)-way
space which as it were surrounds any given 7i-way space.
When we place the suffix b outside a symbol this will
indicate that the symbol belongs to the higher space. The
suffixes will always be l...n. When we have to consider
the suffix which should correspond to the variable u it will
be denoted by a dot.
Let * ^ = _ 2 /2< fe f (146.3)
We see that
(ikh) b = (Usk) a ; (ik -) b = fl ilt tj>; (i- k) b = - fl ik $
{ikh} b = {ikh} a ;
(rkhi),, =
A USEFUL REFERENCE IN (n 4- l)-WAY SPACE 207
in tensor notation,
147. Geometry of the functions /2^. If we are given
any n-way space we shall see that it may be surrounded by
what is called an (71+!)- way Einstein space. We shall
define this space shortly. A particular n-way space may be
surrounded by a Euclidean space of n + 1 dimensions ; but
before wo consider particular kinds of this (714- l)-way space
we had better consider the geometrical meaning of the
functions /2 t ^ which together with (f> are to generate the space.
With this end in view let us consider two geodesies going
out from the same point x l ... oc n , u = 0, and having the same
direction cosines *... n , at this point, the first geodesic
being in the 7i-way space denoted by a, and the second in the
(n+ l)-way space denoted by 6.
We have (140.5) for the current coordinates on these
geodesies x\ ...x' n and x'\ ... x" n
and therefore, neglecting terms of the third order in the arc s,
we see that the coordinates are the same for the two geodesies.
But for the first geodesic the coordinate u is zero, and for
the second geodesic
The distance between two points, one on each of the geodesies,
is therefore 2 '
208 DIFFEBENTIAL GEOMETRY IN n-WAY SPACE
8 being the distance of each point measured along its own
geodesic from the initial point.
This mutual distance we may consider to be normal to the
geodesies if we neglect terms of the third order in the arc.
The curvature of the first geodesic is defined by Voss as
the ratio of twice this distance to s 2 . This is obviously
a proper definition, agreeing with the ordinary definition
when we are dealing with Euclidean space.
We therefore have
^L\^ax x ax^ /1 . O x
m - llTt* . l
^f/'^A^A*
The curvature of the first geodesies may be called the
normal curvature in the (tt,+ l)-way space of the surface
u= in the direction dx lt dx 2 , ...dx n .
Looked at in this way we may write our formula
1 = A/jx^ t (147.3)
jR ^Au^A M
To get what we may call the directions of principal curvature
we require the directions dx t which make ^ critical.
The directions of principal curvature are therefore given by
(a x/i -H/2 A/l )cte /t = O f (H7.4)
where the values of R, the principal radii of curvature, are
given by the determinant
|a A/4 -JB/2 AM | = 0. (147.5)
We shall.now show that the directions of principal curvature
are in general mutually orthogonal.
At any given point we can choose the coordinates so that
corresponding to the principal radius R^ only the coordinate
X} varies at the given point. We therefore have at the point
' a ilt = Ri.fl'j..
IK K IK'
GEOMETRY OF THE FUNCTIONS
209
If, then, all the radii of curvature arc distinct,
a ik = n ik = 0, if i&k, (147.6)
and therefore the coordinates are mutually orthogonal at the
given point: that is, the lines of curvature are in general
mutually orthogonal.
148. The sum of the products of two principal curvatures
at a point. We now wish to obtain an extension of the
well-known formula of Gauss
1 __ (1212)
for a surface lying in ordinary Euclidean space.
Consider any determinant
a
a
nl nn
and the corresponding determinant
If \a\ denotes the first determinant we see that any of its
minors is equal to | a multiplied by the complementary
minor of the second determinant.
Expanding the determinant (147 . 5), or say
we see that the determinant divided by \a\ is equal to
x x 2
'* .*/ i . '* / v! Ill' t'li i'7 / / / / / \
a (a ? *a / ' A -a ?/ 'rt f ^(tt H a ?fk -a rh a i k )-...,
(148. I)
the numerical factors T-TT,- > TT2 "TV*" being introduced
(1!) (2!) (3!)
in accordance with the convention about repeated factors.
We therefore have for the principal curvatures
(148.2)
^-a,;AJ. (148.3)
wit
* K
e
210 DIFFERENTIAL GEOMETRY IN Vi-WAY SPACE
Now consider the expression, a tensor component clearly,
and similarly B ri == b kh (rlchi).
We have seen in 146 that
From the last equation, we see that
Vy /2 rA /V (148 . 4)
.,
From the second equation, we see that
(B,.) b = a'<>><l> (n fh . k -n hk . r ). (148 . 5)
From the first and last equation, that
^. (H8.6)
The expression a ri A r ^ is an invariant which we denote by A.
We thus obtain
or, interchanging i and h,
E = A + a'*a*' (/2, A /2 i7 ,-/2
and therefore
2J5 = 2A+(a r *a 1sh -a rh u ki ) (Sl rh Sl ik -
(148.7)
It follows that
2B= 2A + 4S-~ L jy +40~ 2 5... (148.8)
Kemembering that B is an invariant in the (n+ l)-way space,
and B.. a tensor component in this space, and that A is an
invariant in the ?i-way space obtained as a section of the
(7i+l)-way space by the surface u = 0, we may express
the result at which we have arrived in the following way.
SUM OF PRODUCTS OF TWO PRINCIPAL CURVATURES 211
Consider any (>i-h l)-way space, and a section of it by any
surface.
Let g l ... n ~ Hl be the direction cosines of this surface, re-
garded as a locus in the (n+ l)-way space.
The diiection cosines are connected by the identical equation
1 = !>,#?
The expression V ljt (rkld)^ r g l is an invariant of the
(ft + l)-wuy and the surface we have chosen. When we
take the ground form ^du^-i-b^dx^dx^ and the surface
u = the expression becomes 0~ 2 ./?.., since * ... g n are zero,
and 02 71+1 7* +1 __ i
If B is the invariant of the (n+ l)-way space,
and if A is the corresponding invariant of the /i-way space
which is the section of the (71+!)- way space by the given
surface, then what we have proved is the following.
The sum of the products, two at a time, of the reciprocals
of the principal radii of curvature of the surface, regarded as
a locus in the (>i-f l)-way space, is equal to
-I (B-A)-b l(h (rkhi) b g r tf. (148 . 9)
149. Einstein space. Suppose, now, that instead of
taking any surface we choose a surface whose direction
cosines satisfy the equation
%J3=b Ut (rkhi)g r g ! ', (149.1)
we shall have 1
i^ + J 7r/ ^=0. (149.2)
^i^k
For the case n = 2, this becomes Gauss's well-known
formula t (1212)
li l ^ 2 a n a 22 a j 2
If, then, the (?i-f l)-way space is to be such that for all
surfaces lying in it the formula of Gauss will hold, the equation
i^ = **?'
must be identical with 1 =
212 DIFFERENTIAL GEOMETRY IN n-WAY SPACE
We must therefore have
B H = Wb H , (149.3)
and in consequence of this
that is, B = %(n+l)B. (149.4)
Leaving aside the case when n = 1, we must have
5=0, (149.5)
and therefore B ri = 0. (149.6)
A space with this property is what is called an Ein.stein
space.
It is interesting to see how from mere considerations of
purely geometrical ideas we should be led to Einstein space.
150. An (H+ l)-way Einstein space surrounds any given
91-way space ( 150-4). We shall now show how, being
given the ground form of any ?i-way space, we may obtain
the ground form of a surrounding (n+ l)-way Einstein space.
We look on a ik . t . as functions of x l ...x n and u whose
values are known when it = 0. We have if possible to
determine functions /2 2 - fe ... and <p which will satisfy the
equations a lh (fl tll . h ~fl hl .. t ) = 0, (150 . 1)
(150.2)
6 = -2/2^0, (150.4)
when u ~ 0.
If we can find such functions, we take
and thus find the Einstein form
2 cZu 2 + bftdxidxk (1 50 . 5)
in the immediate neighbourhood of the surface u = 0, that
is, in the immediate neighbourhood of the given n-way space,
DETERMINATION OF SURROUNDING EINSTEIN SPACE 213
and proceeding thus by the method of infinitesimal stages we
ultimately obtain the Einstein space which we require.
Let A^ = a xt {tppn}. (150.6)
We shall first prove the fundamental identity
Employing the geodesic coordinates at a given point,
and therefore in this system of coordinates we have at the
given point
(150 ' 8)
\a >A
Similarly wo have
ft A
and therefore A..
= 0. (150.7)
The required identity holds therefore universally, since it
is a tensor equation which vanishes for geodesic coordinates.
151. We now transform the functions f2 ijt ..., which are
to be found thus. *
214 DIFFERENTIAL GEOMETRY IN n-WAY SPACE
Let V* = a kt n it ,
and therefore fl il{ = ft ki = a^V* = a a F x ,
Wilting F for F we now have
x=V, (151.2)
that is, multiplying by a A ^ and summing,
#(^+
The third equation is
and by aid of the second equation this may be replaced by
A = FF-F 2 . (151.4)
We thus have the equations
V* V A 1/7* y x V2
K A*A %> ^ K A ^A 1 "" K >
and, writing <fy for a x *0 ^ in the other equation (151 . 3),
Tl:ese are only assumed to hold when u = 0.
152. Noticing that = A 2 (0),
we have $. x = Aa x< 0. Mf = a xt \p.<f> t
and therefore ^-A+'^^A ^^^2(0)- ( 152 -
DETERMINATION OF SURROUNDING EINSTEIN SPACE 215
We now operate with A on the equation
where A = + {t\8}(()-{p\t} (<),
Ox \
s being the upper integer here A itself so that we employ
another integer 8 and p. the lower integer in V^.
We easily verify that
A-~ = Ax-(')jLj$x8) +().-
<ht <)', w du ' J V c)u
and therefore
.,. (152.2)
Now, since (4 . 3) x/1 a^ s = ej-,
we see that . - a^ =2<j> a" Vf, (152.3)
V Li
and ~{rkh] = ~a
we see that
*. (152.4)
Since
. (152.5)
216 DIFFERENTIAL GEOMETRY IN n-WAY SPACE
Now if we multiply each side of this equation by 7l and
sum, remembering that
we get
Vl
= v 1 ;. (t k v f + Ff . ,) - 0,. n vl - <f>, : V;. vi
Fl'.,+ F!'.<)
(152.6)
We deduce from the equation for -- {rkh} that
*~ { rklcl = <t >/ V' l . + <t>V l r . t -<t> l .V-<
= -/>, F-0F,. - (152.7)
Combining the formulae \ve have proved, we now see that
*- v^ - v h , (<t> vl ., + ^ v I) - 7; (
We also have - V = </> (A + V 2 ) -f A 2 (0),
o!6
and therefore
A F a = M (4 + V' 2 )
so that r- - (1
A -^-2FF^. (152.9)
Now
and therefore ^ M + 2 FF^, = 2 7* V'l . M .
We have proved (150.7) that 4* . K = %A^
and we have K. A ='F /J ;
DETERMINATION OF SURROUNDING EINSTEIN SPACE 217
it follows that we have
^(V^-VJ = 0. (152.10)
153. Now A is an invariant for any transformation of the
coordinates x 1 ...x n in the ^-way space, and so therefore is
j-~. We shall therefore, in finding an expression for -->
employ geodesic coordinates at a given point and thus materially
simplify the necessary algebra.
and therefore
and we proved in the last article (152 . 7) that
Hence, in geodesic coordinates, - - is equal to
'
Now
218 DIFFERENTIAL GEOMETRY IN n-\VAY SPACE
and therefore
Xt) V 1 ,,
%0
(153.2)
It follows that - is equal to
, (F) + 24'/ F,>. (153 . 3)
Again, a^r^'T*, '
and therefore
^^ n-x = 0/a^ T^., = a">, F^ = A (0, TO,
^^
so that -r is equal to
%u ^
-2FA 2 (0) + 2^r^ + 2^F> (153.4)
Since, then, the equation
^F^ (153.5)
is expressed in invariant form, it is true not merely at the
point, whose geodesic coordinates we employed, but universally.
Now
so that (F^F^-F 2 -vl)= 0. (153.6)
SUMMARY 219
154. If, then, we are given any 7i-way space, and we find
functions V\ such that
which will satisfy tho equations
^x=^, A=VlV*-V, (154.2)
and if, taking arbitrarily any function <p of # 15 J\ 2 , ... x n and
a new variable u t we allow a^. and Vl to grow in accordance
with the laws
^ = -20^ (154.3)
^ = <j>(A i l + VV'' i ) + a'<j>. H , (154.4)
<H(/
taking as their initial values when n the given values ot
a ik in terms of ^ ...#, and the values initially found for Vf,
the equations V ^.^=V^ A = V*V*-V* (154 . 2)
will remain true when u takes any value \vhatever, and the
form 2 c?u 2 + a {u dx { dx u (154.5)
will be the ground form of an (u-f l)-way Einstein space.
The surfaces u = constant may be any whatever in the
Einstein space; and we see (149 . 2) that the property of this
space is that the sum of the products two at a time of the
reciprocals of the principal radii of curvature of any surface in
this space is equal to \ A , where A refers to the n- way space
given as the section by the surface of the (ti + l)-way Einstein
space.
CHAPTER XII
THE GENERATION OF AN (w + l)-WAY STATIONARY
EINSTEIN SPACE FROM AN oi-WAY SPACE
155. Conditions that the (>i+l)-way Einstein space
surrounding a given )i-way space be stationary. Wo have
shown tbat any rt-way space is surrounded by an (n-f-l)-way
Einstein space, and that the equations which lead to the
Einstein space are
V K V' A V x V^ V 2
Y ' > " '
where A* = a xt {t
The Einstein space has the ground form
<t>' 2 du 2 + bji.dxidxj.,
where bjj s is equal to a^ when u = 0.
We now inquire what properties the n-way space ground
form must have if (f) and by c are to be independent of u.
Clearly the necessary and sufficient conditions are that
V\ = 0. (155.1)
Wo therefore have -4 = 0, <J)A^ + 0^ = ;
that is, 0{X/}+0- x ^=0; A 2 (0) = 0. (155.2)
We now 'want to transform the ground form
a ik dx i dx k
and the function (f> by the transformation
<*=&*". <^= (2 - n)r - 055.3)
We have a"- = e~ zv l ik ,
THE APPROPRIATE W-WAY FORM 221
Let* 0'*
where QJ =
and therefore 6- a -ft.
We also have A (0) = 0% A 2 (0) = 0'.,.
It is easy to verify the following relations :
0r,./,-0,Vi = (^-2)d. th + a rh \(0). (155.4)
Now W}a={^'h+^,
and
{rkih} = ^- {rifc} - .-- { r/Jj} + {r} [thk] - |r/^] {fit},
d^ A MI
and therefore
We also sec that
) a -(n-2) (A
and therefore
(A 2 (F)-(ft-2)A(F)) rt -(A 2 (F)) 6 =0. (155.0)
NOW F;,. /( -F,V,+F!.,F;,-F;,,F,,
= (n-2)(F. rA +F r F 7i ) + a,. 7( (A 2 (F)-Oi-2)AF),
and therefore
{rtth} a = [rtth} b + (n-2)(V. rh +V r V k ). (155.7)
It follows that, since
-1) V r V h = 0. (155.8)
* [This is not the introduction of some new function 0, but an assign-
ment of meanings to 0^. and 0* in connexion with any known 0^. The
meaning of f^ is that assigned in ^.] *
222 AN (>l+l)-WAY STATIONARY EINSTEIN SPACE
For the form b^dx^dx^ we therefore have
A 2 (F) = 0, {rtth} b + (n-l)(n-2)V r V h = Q. (155.9)
If we can find a ground form to satisfy these conditions, then
<* = & Sr . = e( 2 -"> r (155.10)
will give a ground form which will lead to an (7i + l)-way
Einstein form, the coefficients of which will be independent of u.
158. Infinitesimal generation of the (?i-fl)-way from
the n- way form. In order to simplify the problem of finding
the ground form b^dx^lx^ we shall regard it as an (n+ 1)-
way form and bring it to the form
* <p*du~ + bfadzidXk , J = 1 . . . n, (156.1)
as we have done before.
The equations
A, (V) = ; {rtth} + it (n- 1) V r V h =
now become, if we take V Vn ( 1 n) u,
{rtt-} = 0. (156. 2)
If we regard the form as generated from ^ i^dx^Lr^ wo
have ( 146) the equations
We therefore have
., i - ^n^-a^a^n^n^, (ise . 3)
= a"'(n r!t . k -n Mi . l ), (156.4)
o = 4>(A H + u Ki'(n ri n hk -n rK n i d-
*
The tf used here is not the ^ of the Einstein form.
GENERATION OF THE (tt-f l)-WAY FORM 223
The first equation may be replaced by
- 0- 2 = A + a^ (fl fi fl^ - fl,^ a* 1 .
The geometrical meaning of this equation, since it may be
written
is
12
that is, the sum of the products two at a time of the principal
curvatures of the equipotential surface V= regarded as
a locus in the (n+ l)-way space is equal to %(A + 0~ 2 ).
Making the transformation to the functions VI we have
^=-20a f . v y?, (156.7)
and, from [A 2 (u)] b = 0, we have
that is, - X/1 /2 A A-' -0- 4 ^ = 0,
16
or - F0- 1 - </>-> ^ = 0;
c)U
so that ^ + F0 2 = 0. (156.8)
We have, a denoting the determinant of the form a ili dx /i dx k ,
t>ct j*ba
- - = aa l
- - -
en du
= - 2 aa i/f f2 ih (f) = -2a V<f> ;
and therefore ^ (a0~ 2 ) = -2aF0- 1 -2a0~ 3 ^;
o '\jj v 16
so that the equation -~ + F0 2 =
may be replaced by (<^0~ 2 ) =- 0. (156. 9)
224 AN (>l + l)-WAY STATIONARY EINSTEIN SPACE
We have the equations
As earlier, we therefore see that
(156.10)
and this ia equal to zero, since
0- 3 <^ = i^-J 7 X>+ FF x-
We also have, from what we proved earlier (153 . 5),
and we have
i. (0-2 + F 2 - 7$; F^) = - 2 tfr 3 1| + 2 F (0 (vl + F 2 ) + A.
and therefore (^ + 0~ 2 + F 2 - F
) = 0. (156.11)
We thus see that the required (7i,+ l)-way form can be
generated from any w-way form innnitesimally by choosing Fl
and to satisfy the equations
F* =F,,; A + 4>-*+V*-V*V$ = Q. (156.12)
RESTATEMENT AND INTERPRETATION OF RESULTS 225
157. Kestatement and interpretation of results. We
may now restate the result at which we have arrived.
Let a^dx^dx^ be the ground form of any 7i-way space
whatever.
Find functions Ff such that a^ K V i = a iK V k , and which
satisfy the equations F*. x = T^.
Define a function by the equation
where A=a
Let the coefficients a^ and the functions and Ff grow,
with respect to a new variable u, according to the laws
having as initial values, when u = 0, the given values of a^ c
in terms of x^ ... a; M and the values initially found for F* and (/>.
The equations I/A __ rr
H y - y
will remain true when u takes any value whatever ; and,
a denoting the determinant of t^..., a</>~ 2 will remain
a function of x l ... x n only.
Tho (n+ l)-way form
(/> 2 du 2 + a^d.^dxj., i = 1 . . . ?i,
will now have the properties
A 2 (u) = 0; {.ft-} = l; {rA}=0; {r^-*} = 0.
Transform now to any new variables which we may still
denote by x l ... x n , % n+l , and let
n (n 1) =
and thus let the (71+ l)-way form be
226 AN (^+l)-WAY STATIONARY EINSTEIN SPACE
It will now have the properties
where A 2 (7) = 0.
From this (n+ l)-way form let us pass to the form
bftdxidxfr
where b ik = a ih e 2v y
and let 6 = ^ l ^ v .
We now have, for the (u-f l)-way form
Iwlxtdxk, [ = 1 ...71+1,
or = 0,
where ^ = ft*' {rtf/ij, = 6^ (9.,^.
The (?n-2)-way form Q^du^-^b^dx^lx^ will now bo an
Einstein form and the coefficients b^ and 6 will he inde-
pendent of 16.
158. A particular ease examined when n = 2. As
a particular case we might consider what properties the
n- way space must have if in the (u + l)-\vay form which it
generates, namely 02</ w 2 + a^.d^d^. , (158.1)
the coefficients a^. and the function are to be independent
of u.
We must have, as the necessary and sufficient conditions,
A -4 0" 2 - 0,
0^ + ^'0. //t =0,
that is, A H- 0~ 2 0,
0(A^}+0. ;At = 0. (158.2)
Now the chief interest of an Einstein space is when its
dimension is 4. We shall therefore only consider this special
case when n = 2. We thus have
that is, '20 2 A r r= 1. (158.3)
A PARTICULAR CASE EXAMINED WHEN U = 2 227
The other equations become
^a ll A"-0. 1l = 0; 0a 12 /t r -0. ]2 =0; 22 tf-0. 22 = 0. *
(158.4)
We wish to find the properties of the two-way form which
will satisfy these tensor equations.
The element of length on the corresponding surface* we
take to be given by ds * = 2e - e dudv.
We then have {111} + ^ = 0, {222}+0 2 = 0,
{112} = [121} = {122} = {221} = 0,
and A" = e*0 12 .
The equations which have to be satisfied will now be
0,i + 0i0i = 0; 200 I2 = C -'; 23 +0 a 2 =0;
20V0 ]2 = 1. (158.5)
We should notice that the suflixes in these differential equations
denote ordinary differentiation, and not tensor derivation
which would be indicated by the dot before the suffixes.
By means of the equation 200 12 = e~~ wo can eliminate
from the other three equations, and we see that they reduce
to the two equations
00i0ii2 = 0i2(00ii-0l)>
0020221 = 012 (0022-01)- (158.6)
These two equations may be written
^- ( lo 0i2 + l<>g -log 0,) = 0.
Consequently *' 20 and ^ 20 (158.7)
01 02
are respectively functions of v only and of it only.
Wo do not IOFO in generality by assuming that
0J20 = 01 = 02
and therefore = F (u + v)\
where F"J? = P .
228 AN (tt + l)-WAY STATIONAKY EINSTEIN SPACE
The ground form of the surface is therefore
<fc a = iFdudv, (158.8)
where F is a function of u + v given by F" F = F'.
If we take the parameters on the surface
then ck 2 = F' (x) (dx 2 + dy*), (158.9)
and F" (x) F(x) = F' (x). (158 . 10)
The surface is thus a particular case of a Liouville surface.
159. General procedure in looking for a four-way
stationary Einstein space. In general, when we want a four-
way Einstein space of the form
6 2 du 2 + b ik dx i dx k , j. = 1, 2, 3,
in which the coefficients b ik and are to be independent of u,
that is, what is called the ' stationary ' form, we begin with
the ground form ail dx\ + ^ a^dx^x^ a^dx\ . (159 . 1)
We then find in any way three functions
/2 11S /2 12 =r /2 21 , /2 22
of the parameters x } and # 2 which satisfy the tensor equations
/2 n . 8 = /2 ls . lf /2 2ri = /2 12 . 2 . (159.2)
We define a function by the equation
/2 n /2 22 + /2? 3 . (159.3)
We now let the coefficients a ik and the functions fi ij grow
with respect to a new variable u in accordance with the laws
7J? = - 2 * fl tt' < 159 - 4 )
75?= *' ifc - 15
The equation which defines <f> will be unaltered, and the
equations /2 U . 2 = B^^, ^^ = B M . g (159. 2)
will remain true. * r
QUEST OF FOUR-WAY STATIONARY EINSTEIN SPACE 229
We thus attain the three-way form
4> 2 cZu 2 + a n dx\ -{2a^ 2 dx l dx 2 + a^dx\, (1 59 . 6)
in which in general </> and the coefficients a^ will be functions
of #!, # 2 , and u.
In the particular case we considered in the last article,
when the two-way form appertained to a particular class of
Liouville surface, the function </> and the coefficients a^ will
not involve u.
But in all cases the three-way space with the form
will have the property A 2 (u) ~ 0, (159 . 7)
{ittk} = 0, {itt-} = 0, {} = !, (159.8)
and therefore
(2323) = -fa 22 ; (31 31) - _ * ; (31 23) - Ja 12 ;
(23 12) - ; (31 12) - ; (12 12) = "
(159.9)
From this three-way form we can deduce the ground form
of a stationary four- way space by the rules we have given in
the general case.
We should notice that if we begin with the proper Liouville
surface, the Einstein stationary form at which we arrive
can be, by a proper choice of the parameters, thrown into
a form in which all the coefficients will be functions of two
parameters only.
160. Conclusions as to curvature. The three-way space
with the form
is such that, if we regard the surface u = constant as lying in
it, the product of the reciprocals of what we have called its
principal radii of curvature is
J/11 UoO """"" U T 9
11 22 - f 1 - (160,])
a ii a 22~~ a f2 *
We must not confuse thgse racHi of curvature with the
230 AN (n+l)-WAY STATIONARY EINSTEIN SPACE
radii of curvature of the surface n = constant regarded (as it
may be) as lying in Euclidean space.
We have in fact the theorem : * the product of the reciprocals
of the principal radii of curvature of the surface, u constant,
is equal to ]{ --J-0-*, (160 . 2)
\vhere K is Gauss's measure of curvature/
Riemann's measure of curvature corresponding to the
vectors and 77 which lie in the tangential Euclidean space
at any point is ( 141)
{]GO 3)
i , l i i )'
where the direction cosines of the vectors and 77 are re-
spectively \ 2 , 3 , and 7/ 1 , r; 2 , rj''.
If the vcctois and TJ touch the surface u = constant, this
becomes -|0~ 2 . If their plane contains the normal to the
surface it becomes \ " 2 .
In the particular case when we start with the proper
Liouville surface these are respectively K and K, where K
is Gauss's measure of curvature.
CHAPTER XIII
it-WAY SPAOK OF CONSTANT CURVATURE
161. Ground form for a space of zero Rlemann curvature.
We shall now consider the simplest form in which the ground
form of a space may be expressed in which Riemann's
measure of curvature is zero everywhere and for all orientations.
For such a space {trpr/} (161 . 1)
for all values of the integers.
Consider the system of differential equations
0.^ = 0. (101.2)
We have 0.^,. = 0, <f>. prq = 0,
and therefore (* <] <]?') 0,, >
that is, tlrfrq] fa = 0. (161 .3)
A system of equations with the property that no equation
of lower order can be deduced from them by the processes of
algebra and of the differential calculus is said to be ' complete '.
The necessary and sufficient conditions that the system of
differential equations 0.^ = may be complete is then
-\ptrq] = 0, (161 . 1)
that is, that Riemann's measure of curvature is everywhere
zero.
If 'ii and v are any two integrals of the complete system
-0, (161.4)
and therefore, since A (?(, v) is an invariant, it is a mere
constant.
232 ?i-WAY SPACE OF CONSTANT CURVATURE
If, then, we take any n independent integrals of the equation
system 0.pg = (161.2)
as our new variables, the ground form will take such a form
that each a^ is a mere constant.
The ground form can therefore be so chosen as to have the
Euclidean form d*l + ...+ds* m (161 .5)
162. Ground form for a space of constant curvature for
all orientations. We next consider the ground form which
corresponds to a space for which Rieinann's measure of
curvature is the same constant for all orientations.
We have (Xw) = K (a^a,, p -a^ V ),
ami therefore [Xtpp} = Kfr^-a^fy. (162.1)
If, then, t ^ p and t ^ (a,
we Lave :X/z/jj=0, (162.2)
and (A2W}=A'a AM " (162.3)
it p is not equal to p,. Here the repeated integer p is not to
have the usual implication of summation.
Consider now the system of equations
u ftq + Ka l)q u=0. (162.4)
We see that tho system is complete : for
u. tr + Ka )q u r = 0,
r u q = 0,
and therefore \ptrq] u t + K(a ljq u r a pr u^ = 0,
and for a space such as we are considering this is a mere
identity and not a differential equation of the first order.
The system is therefore complete.
Now let u be any integral of the complete system (162 . 4).
We have ? & u = fa^
= - K (
CONSTANT CURVATURE FOR ALL ORIENTATIONS 233
and therefore Au -h Ku 2 (162.5)
is a mere constant.
As u does not satisfy any equation of the first order, being
defined as any solution of the complete system, we can choose
ib so that A (u) + Ku 2 is eipial to zero at the origin, and
therefore zero everywhere.
Let A r = - j (102. G)
We now have A (log u) = />2
Take now as now variables
where A (//,, 7/ 7 .) = 0, & = 2, ... u.
The ground form will take the form
dx'[ -f a^dx^dxj.y * 2, ... a,
i r/ /"/'"'
ami, since '^-^r/ "TJ-T"
and u
we have ! tk 1 \ H j~- = ;
that is, (ikl) + - = 0,
t> 2
or - (log ay.) -p
It follows that a i1f = e" />,-/., (102 . 7)
l/t t/l. 7 \ /
where b^ is a function of .i' 2 ... x n only.
As regards the form b.^dx-dxi., ' = 2, ... ?,,
wo see that, since
' 2x i
(rldh) a = e"^ (rkih) b + {ikl} (rhl) - {hkl } (ril)
H L
234 U WAY SPACE OF CONSTANT CURVATURE
and since (162.1)
wo must have (rkih)^ = 0. (162 . 8)
It follows that the ground form of a space of constant
negative curvature may be taken as
dxl+e (tteS + ...+<&,;). (162.9)
By the substitution
the form may be written
7) >
(102. 10)
The corresponding form for a space of constant positive
curvature may be taken as
7?2
f -(-<Lx\+dx\+...+dxl). (162.11)
Xi
163. Different forms for these spaces. We may find
other forms for these spaces.
Taking the case of positive curvature, instead of choosing u
so that A(u)+ Ku* = 0,
we may choose u so that
(163.1)
/>
Let u = cos z ,
then A (x^ 1,
and the ground form may be taken
dx\ + a ik djCidx k , [ = 2, ... n. (163.2)
Since u
we now have (i/ol) + %^ cot ^J = 0,
DIFFERENT FORMS FOR THE SPACES 235
and therefore a v . = sin 2 ^k- ; ., (163 3)
1 JLi
where b^, depends on #' 2 ... x n only.
We have
(ikl} (rhl)-{hkl} (ril),
and therefore
rf^ = ;* ^ (rA tt )6 + cot* 5 " ' *"H^4^ ;
that is, (,*;/o,. = *''**-> V
The ground form may therefore be written
where b^dx^dxj. is a ground form in o^...^ only, with the
same constant positive measure of curvature.
It at once follows that the ground form for a space of
constant positive curvature may be written
/-V> /Y*
dx\ + sin 2 '! tU-r t + sin 14 -.! sin 2 '-J dx\ + . . . ;
/i li JK
or perhaps better as
7i 2 (c/a;? 4-siir a.y/.r? -|- sin-a. t 1 sin 2 a i 2 (/iC5 -f ...). (1G3 . 5)
A form obviously equivalent would be
li 2 (dx* + cos 2 a\dx^ + cos 2 a^cos 2 xjlx\ + ...) (163.6)
The latter form when applied to a space of constant negative
curvature would become
-R* (dx\ + cos 2 ^^^ 2 + cos 2 a\ cos 2 X^IJL\ + ...), (163.7)
and this may be written
R*(dxl +cosh 2 .r l (te| + cosh 2 ^ cosh 2 x^dx\ + ...). (163 . 8)
The surface ^ = constant, that is, the (n l)-way space
x. z ... x n , regarded as a locus in the oi-wny space of constant
curvature given by the form
(ta| + ..., (1G3.9)
236 71- WAY SPACE OF CONSTANT CUKVATURE
has all its principal radii of curvature equal to
n x*
H tan -73 9
M
and any line on the surface is a line of principal curvature.
164. Geodesic geometry for a space of curvature -f-1.
We shall now consider the geodesic geometry of a space
whose curvature is positive unity : that is, the space corre-
sponding to the form
dt? = d-jci -f sin 2 aj,^? +sin 2 aJ 1 sin 2 a3 J rfo; + .... (164 . 1)
We shall first find the equation which a path must satisfy
if it is to he stationary with respect to variation of the
coordinate x l .
If we write for -4-* we must have
* <is
and therefore, since
1 = x\ -f- sin 2 x v x\ -f sin 2 #, sin 2 c. 2 ? t + >
wo have ^ = cot^ (1 -x\). (1G4.2)
It follows that - l . sin 2 ;r x (x\ - 1 ) = 0,
and therefore cos j\ cos a x cos (s + l ) 1 (1G4 . 3)
where a a and l are constants, and s is the arc measured from
some point on the path.
It follows that
_- - - - r/
(1 COS'' Ofj COS^ ( -I- a ))"
-, tan (s 4- f,)
l -----
(164.4)
T ^ A -
Let **, = tan
1
sin
then we have ds'i = rfe^ -f- sin 2 # 2 cZ#o + . . . . (164.5)
Here 8 l is the arc in, an (/i-l)-way space of curvature
positive unity, and if 8 is to be stationary for variation of # a ,
then 8 1 must also be stationary. '
GEODESIC GEOMETRY FOR A SPACE OF CURVATURE + 1 237
Proceeding thus we see that the equations which define
a geodesic are CO s x l = cos a t cos ( + e^,
cos ,T 2 = cos a 2 cos^ + 6 2 ), . . . , cosu^j = cos a n _ 1 cos (s n _ 2 4- e n _i)>
a 'n == 8 n-l + n>
sin a t tan s 1 = tan (s -f fj), sin or 2 tan s 2 = tan fa + e 2 ), . . . ,
sino^.jtans^ = tan ( /4 _ 2 4- e M _,). (1G4.6)
If we take
j r cos a? x , 2 = r sin a\ cos i^ 2 , , r sin ^j sin o? 2 cos C :i ,
n = rsinaj 1 ... sina' 7? _ 1 cos,r ?i , w+1
we see that ! +I + ."+n+
and we easily verify that
(164.7)
The 7i- way space of curvature positive unity is then the
section of an (?i + l)-way Euclidean space by a sphere of
radius unity.
165, Geodesies as circles. We shall now prove that
every geodesic is a circle of unit radius in ordinary Euclidean
space of three dimensions, but generally two geodesies will
not lie in the same Euclidean three-fold.
We have for a geodesic
sin x r cos s r sin a,, cos (^-.j 4- e y .), sin x r sin s r = sin (s r _ l + e r ),
and therefore
sin x r cos x r+l = A f cos s r _ 1 + B f sin ts r ^ l
= a r cos( r _ 1 + J .) + 6,.sin(8 r _ 1 + r ), (165. 1)
where -4 r , B r , a r , &,. are some constants.
It follows that
sin cc r sin x r+1 cos cc,. +2
= sin a 1 /. (-4,. +1 cos 8 r + B r+l sin s r )
= J. r+1 sin a r cos (8,..! 4- e r ) + J5 y . +1 sin ^..j + e r ),
and therefore
b r cos a r sin x r sin x r+l cos ^ r4 . 2 fe r+1 cos a r sin j' r cos a? r4 1
= (JB r+1 (f r sina r 6 r -4 r+1 )cosa: l .. (165.2)
238 tt-WAY SPACE OF CONSTANT CURVATURE
We thus have a linear relation between the three coordinates
/> s/'+l' br+2*
By a linear transformation in the ('Ji-H)-way Euclidean
space t ... w+1 wo can take it that the first such relation is
1 = ' and that ? +j + ... +* + i = *
Proceeding thus with respect to any one geodesic we can
take it that the equations which define it are
i.1 1
that is, ^ t = , ar a = , ... a? M-1 =
t t &
It is therefore just a circle in the space given by
'' 2 = d+'*!5 + i. 065 - 3 )
and its equation is j{ +^- + 1 1, (165 . 4)
with & = 0, & = 0, ... _, = ().
166. Geodesic distance between two points. We shall
now find an expression for the geodesic distahce between any
two points in the u-way space whose measure of curvature is
positive unity.
Let the two points whose coordinates are
a, ...#,* and y^...y n
be denoted by x and y y and cousider the geodesic which joins
the two points. Let s, s lf ... ,s /l _ 1 be the arcs which correspond
to x, and s', s' 19 ... s'^-i the arcs which correspond to y.
We have
cos x l cos 2/ t + sin x l sin y L cos 5 t cos ft\ + sin ^ sin y l sin ,_ sin s\
= cos 2 a, cos (s 4- e^ cos (s' -f fj) + sin 2 j cos (s + 6j) cos (' 4- 6j)
+ sin (s -t- 6 t ) sin (' + l ) t
and therefore
cos(s' s) = cos x l cos 2/j 4- sin x l sin y t cos (8\ s l ). (1GG . 1)
Similarly we see that
cos^'j si) = cos cc 2 cos 7/ 2 4- sina? 2 sin y cos (8' 2 sj (166 . 2)
(1C6.3)
GEODESIC DISTANCE BETWEEN TWO POINTS 239
It follows that, denoting the geodesic distance between the
points x and y by (xy),
cos (xy) cos x l cos y l + sin x l sin y l cos cc 2 cos y >2
+ sin 0,'j sin y l sin # 2 sin y 2 cos # 3 cos y., 4- . . .
+ sin x l sin ^ sin x^ sin ^ ... sin j? M-1 siii y ll _ 1 cos (x n -y n ).
(166.4)
This is the formula which is fundamental in the metrical
geometry of w-way space of curvature positive unity.
167. Coordinates analogous to polar coordinates. We
can now employ a system of coordinates, to express geo-
metrically the position of any point in our space, which will
be analogous to the use of polar coordinates in ordinary
Euclidean space.
We take any point in the space as origin, that is, the point
from which we are to measure x l , the geodesic distance from
the origin.
It will be convenient to denote this distance by tan" 1 r, so
tliat r = tana: 1 . (167.1)
Let us now consider the system of geodesies which pass
through this origin. For any one of these geodesies # 2 , ... x fl
are fixed, and we may therefore regard *r 2 , ...x n as the co-
ordinates which define the geodesic, and thus regard r, a' 2 , ...x n
as the polar coordinates of a point in our space.
The geodesies through the origin cut the surface r = con-
stant iii an (?i l)-way space of positive curvature 1 +r~ 2 .
In particular the surface r = infinity (167 . 2)
is an (/i l)-way space of curvature positive unity, and the
coordinates of any point in this space define a geodesic through
the origin.
The geodesic distance between two points at smull distances
x l and y 1 from the origin is given by
cos (^2/1) = cos x l cos 2/!
+ sin x l sin y l (cos x 2 cos y 2 + sin x 2 siny 2 cos # 3 cos 2/ 3 4- . . .),
and therefore t
X 2 1/ 2
240 ?-WAY SPACE OF CONSTANT CURVATURE
But, if is the angle between the geodesies through these
points and the origin,
(^2/i) 2 = x l +2/1 - 2 ^i 2/i cos 0-
It follows that
cos Q = cos x 2 cos // ,, -f sin :c sin //,, cos # a cos y 3 -f . . . , (167.3)
that is, the angle between the two geodesies is the geodesic
distance between the points where the geodesies intersect the
surface r = infinity.
The geodesic distance between any two points is therefore
the geodesic distance between two points, on a sphere of unit
radius, whose polar distances from a point on that sphere are
#! and 2/1, and the difference of whose longitudes is the angle
which the geodesies through the points cut out on the surface
r = infinity.
168. The three-way space of curvature -f 1. We now
limit ourselves to the case where n = 3, that is, the three-way
space of curvature positive unity. For this space x is the
geodesic distance from the origin ; and x (2 and x (} may be
taken as the polar coordinates of the point on the two-way
surface of positive curvature unity, ./'! = . where the
<L
geodesic, through the point x, x%, x 3 and the origin, inter-
sects the surface.
We may without loss of generality suppose that x l lies
between and , x. 2 between and TT, and # a between and
i
2n. In the surrounding four- way Euclidean space x will
then always be positive.
Through 'two points in our space one, and only one, geodesic
can be drawn, unless the two points lie on the same geodesic
through the origin, and are the two points where that
geodesic intersects the surface x 1 =
Through three points in the space we can in general draw
one, and only one, two-w&y locu^s of positive curvature unity.
THE THREE-WAY SPACE OF CURVATURE +1 241
We see this by noticing that three points (x l9 ;r 2 , x 3 ), (y l> y 2 , y^),
and (z lt 2 , s :j ) determine the plane
in the surrounding Euclidean space. The exceptional case
would be when the three points lie on the same geodesic.
By a linear transformation, in the Euclidean four-way
space, we may take the plane to be t = and the locus of
the points of intersection with the sphere to be given by
There will then be a corresponding set of coordinates
o/p .r 2 , r, such that tlie locus is given by x l = in the new
*"" u
coordinate system.
It will be convenient to call any^ two-way Jocus of curvature
positive unity a plane, though we should remember that it is
only properly a plane in the Euclidean four-fold. Similarly
we shall call any geodesic a line.
Plane geometry in our space is therefore just spherical
trigonometry.
169. The geometry of the space. We may now introduce
a different system of coordinates in order to bring out the
relationship between the geometry of space of curvature
positive unity and that of ordinary Euclidean space.
Let x tan n\ sin x 2 cos ,r ; , ,
y = tan x { sin .r 2 sin ;r 3 , z tan Jc l cos t r 2 . (169 . l)
In this system of coordinates the geodesic distance between
two points (x, y, z} and (,', y', z') will be
cos" 1 ^-- t -i~^ yri ' " (169 . 2)
where r 2 = x 2 -f y 2 + z*.
The square of the element of length will be given by
dtp = (1 -f r 2 )" 1 (dx* + dy 2 * 4- dz* ( -f r-)~ l r 2 dr-) ; (169 . 3)
but in this geometry, as ir) EucMcan geometry, having the
242 7fc-WAY SPACE OF CONSTANT CURVATURE
expression for the actual distance between any two points, we
do not need to make so much uso of the expression for the
element of length.
The equation of any plane is
\x + p.y + vz + 8 0.
Now a plane, we know, is a two-way surface of curvature
positive unity. Let x lt y v z l be the coordinates of its centre,
that is, the point at a geodesic distance - from every point
u
of it.
We then have sx l + yy { + zz +1=0, (169.4)
and therefore Bx l = X, By l = //, 8z l = v.
The angle between two planes is, as in spherical trigonometry,
the supplement of the angle that is, the geodesic distance
between their centres.
The cosine of the angle between the two planes
,1 r 12 15i , v ,,,
is therefore l - 2 ~~ f ^^\^ l 2 ~ l - - (1 GO 5)
(Xf +f*i+vl + Sir(W+Ltl+i> 2 > + 8rf
The equation of a plane, given in terms of the coordinates
of its centre, is ^ + yy v + ^ + 1 = 0. (1 G9 . 4)
The condition that the plane passes through the origin, that
is, the point where x, y> and z are each zero, is that its centre
should lie on the plane whose centre is the origin.
The equation of a line is given aw the intersection of two
planes tCXl + yy l + ^+1 = 0,
^2 + 2/2/2 + ^2+ 1 0.
In connexion with this line we consider the line joining the
points (x lt y^ X ) and (u? 2 , # 2 , z >2 ).
The plane whose centre is A may be called the polar plane
of A. We see that if J5*lies on the polar plane of A, then A
lies on the polar plane of .&
THE GEOMETRY OF THE SPACE 243
We now see that if (x l9 y^ X ) and (x, 2 , y 25 z 2 ) are any two
points on a line, then every other point on the line is given by
Z = 1* 9 (1G9.6)
J
where p : q is an arbitrary parameter.
The line given as the intersection of the planes
^+1 = 0,
stands therefore to the line joining (x ly y ly z^) and (# 2 , y^ 2 )
in the relationship, that the distance between any point on
the one line and any point on the other line is - . The lines
Li
which are in this relationship will be called polar lines.
We now wish to consider two lines, viz. the line given by
C3 2 +l = 0,
and the line given by
^4 + 2/2/4 + S3* +1 = 0.
If these lines intersect, the four points
('''l i 2/1 . C l) 5 ('<'*> ?/2^2)> (^3 2/3 . -a)> (^ 4 #4> -4)
lie on a plane, and we thus see that if two lines intersect their
polar lines also intersect, and the plane on which they lie is
the polar plane of the point of intersection.
170. Formulae for lines in the space, and an invariant.
Just as in Euclidean geometry, a line has six coordinates.
We define these coordinates
I = # 2 - iH t , m = 7/ a - y l , n = z, -z v X = y^ - y^ ,
The six coordinates are those of the line joining the points
(x l , I/,, , j) and (ic 2 , ?/ 2 , ^ 2 )> an< ^ ^ e y are connected by the relation
ZA + w/*-Hii'= 0. (170.2)
We easily see that if Z, m, w, A, /i, V are the coordinates of
a line, the coordinates of itfj polar line are A, /z, i/, ^, ?>i, n.
244 71- WAY SPACE OF CONSTANT CUKVATURE
If (12) denotes the geodesic distance between the points
COS (12)=- y^y,
' J*
and therefore sin' (12)= m + tt + + j* + . ( }
' (l+^i)(l+^) ^ ^
Consider the expression
IV + w/m' 4- < ?m / + XX" + / + i/^
_
/* 2 + A- + / 2 + i^*(^^
where (Z, m, -H, A, /^, i/) and (/', m', /i', A', /, i/') are the co-
ordinates of the lines which respectively join the points 1 and
2, and the points 3 and 4.
It is easily verified that the numerator of the expression is
and the denominator is
^l 1 + 6 'i) (1 + *y I 1 + 6 '.;) (I + ^4) ^' m (12) sin (34).
The expression is therefore equal to
cos (13) cos (24) -cos (14) cos (23)
sin (12) sin (34) ' (170.4)
and this is clearly an invariant.
Suppose now that the points 1 and 3 coincide. The ex-
pression becomes cos(24) - cos (14) cos (12)
sin "(12) sin (14) '
and we<see that this is the cosine of the angle between the
lines 12 and 14.
Suppose next that the 6 line 13 is perpendicular to the lines
12 and 34.
FORMULAE FOR LINES IN THE SPACE, AND AN INVARIANT 245
Clearly, from the formula
cos (24) = cos (14) cos (12) (170.6)
when the lines 12 and 14 are perpendicular, the line 13 will
be the shortest distance between the lines 12 and 34.
The planes 132 and 134 will be the planes through the
shortest distance and the lines 12 and 34.
We may, to interpret the expression
W + in m' + 'tin' + XX' + fifi + i/i/
since we have seen that it is an invariant, take the points
1, 2, 3, 4 to be
0,0.0; <r 2 , 0, 0; 0, 0, s., ; 4 ,?/ 4 ,c a ;
it now becomes ^ =
The equations of the planes 132 and 134 become respectively
The angle between these planes is
cos" 1 JL_
The shortest distance between the lines is
1
cos~
and therefore the invariant expression is equal to the product
of the cosine of the shortest distance between the lines into
the cosine of the angle between the two planes drawn through
the shortest distance and the two given lines.
The invariant vanishes if the lines are polar lines. It also
vanishes if the planes through the shortest distance and the
two lines are perpendicular.
If the lines are not polar lines and if the invariant vanishes,
we see that the polar line of 12 intersects 34 and the polar
line of 34 intersects 12.
246 ft-WAY SPACE OF CONSTANT CURVATURE
171. Volume in the space. The expression for the element
of volume in a space of three dimensions and with the
measure of curvature positive and equal to unity is
sin 2 x l sin .r^ dx l dx. 2 dx. 3 , (171.1)
returning to the original notation of 1G8.
The volume enclosed by an area of any plane that is,
a two-way surface of curvature positive unity and the lines
joining the origin to the perimeter of the area is
los ojj) sin # 2 (L'2 dx } , (171.2)
where x l is the geodesic distance from the origin to a point
within the perimeter.
If the plane is at a geodesic distance p from the origin we
can use the equation tan p = tan x l cos x. 2 ,
and express the above integral in the form
P X 7
i \\P~~ ^ an P x i c t x i ' ( '''> > (171.3)
J
where x 1 is now the geodesic distance to a point on the peri-
meter from the oiigin.
If we take r to be the geodesic distance of a point on the
perimeter from the foot of the perpendicular, and take x [} to bo
the corresponding longitude 6 in the plane, the above formula
becomes
- p / tan p cos p cos r cos" 1 (cos p cos r) \ A
2 ( /> =-^ 7- )<10- (171 . 4)
J v VI cos p cos" >' /
If the foot of the perpendicular lies within the area, this
formula gives us for the volume the expression
fsin p cos r cos" 1 (cos^ cos r) fi
pTT - ~~ j= -~=^= (6C7, (171 . 5)
J V 1 ~COS Z ^> COS- ?'
where the integral is to be taken round the perimeter.
We notice that in space of curvature positive unity when s,
the variable in the equation of a geodesic, increases by 2 TT,
then 8 1? s 2 , ... also increase by 2 TT, and therefore the coordi-
nates #, , . . . x n all increase t>y 2?r. cWe thus, in proceeding along
VOLUME IN THE SPACE 247
a geodesic, come back to the point we started from. We cannot
have any two points at a greater distance from one another
than TT.
172. An ti-w&y space of constant curvature as a section
of an extended Einstein space. We now wish to consider
the 'ii-way space of constant curvature as a section of an
(/6 + l)-way surrounding space.
We take the (71+ l)-way space ground form to be
pdut + bftdXidx!., I = 1 ...>*, (172.1)
where b ik = a ije when u 0.
We have ( 146)
(r- A-) 6 -
Extending the definition of an Einstein space, wo shall now
say that a space is an Einstein space if
bM(rkih) = cb lh , (172.3)
where c is a constant.
We have
%. (172.4)
If the surrounding (u+l)-way space is to be Einstein
space according to the new definition,* we must have
[Called iii 18 311 extended Einstein space. \
248 W-WAY SPACE OF CONSTANT CURVATURE
ca ri = J rt + * ft (/2 rt J2 4fe -/
That is, if Ff = u kt n tl (172.5)
where /2, 7 , = /2 H = a,, x F* = </, <A F*,
we must have c = $-* (\ ((f>)-a ri ^ fl, ; ) - F* F^ , (1 72 . G)
VJ.,= F M . (172.7)
Now ^^^
and therefore
Multiplying by ct'^ and summing,
ref = A V , + v j ; v + <f>- 1 0J 1 - -- F'; (172 . 8)
We also have
That is, we have
~
We may replace the equation
iV=**<*
by ^ + F 2 - F F^ = (TI - 1) c.
A SECTION OF EXTENDED EINSTEIN SPACE 249
The equations therefore become
A + V 2 -V*V = (n-l)c, (172.9)
F^ = ^ ( (172.10)
= _20 ft y;. (172.12)
We may easily verify that the results which we have
proved for the case c = still hold in this more general
Einstein space.
The special conditions that the coefficients b^ and the
function may be independent of u become
A = (M-l)c, H + = c0,
that is, 0{Afy*} + 0.AM 0tfA/n A 2 = 0. (172.13)
Now let us assume that the ?i-way space is of constant
curvature K. We have
{\ttfi} = (l-n)A r ra/z.
If we choose /i so that //& = -C, the conditions that the
surrounding Einstein space may satisfy the required conditions
become ^ + Ka^ = 0,
Az(<f))+Ku(/) = 0.
The second condition is a consequence of the first set, and we
see that all that we need is that the system
0.A/i + A r a A Ai0 = (172.14)
may be complete.
We know it is, and thus we may take </> = cos# 1} and the
space o'iven by
cfe 2 = cos 2 ^ du z + R- (dx \ + sin 2 ^ dx * + . . . ) (172.15)
will be an Einstein space of the kind required.
If the space is of constant negative curvature we should see
that regarded as a locus in (n 4-1)- way Euclidean space it
would be an imaginary section.
The expression for the geodesic distance in space of negative
curvature unity is given by
cosh s + sinh x 1 sinh y l + cosh x l cosh 2/^sinh # 2 sinh y 2 + , . .
f sinh x l . . . sinh x n _ 2 sinh y l . . . sinh y n _ 2 cosh x n ^^ cosh l' n - l
... cosh x n _pco*>h[y l ... cosh y n - l cosh (% n --y n )'
CHAPTER XIV
vi-WAY SPACE AS A LOCUS IN (n+i)-WAY SPACE
173. A space by which any u-way space may be sur-
rounded. We now consider again the ground form of an
(>b+ l)-way space du* + b ik dx i (lx j . which, when we put u = 0,
becomes a^dXfdx^.
We have, by the formulae of 146,
{Mi},, = {ikh: a - {ik-] b = n {h -, {i-k\ b = -
= * + 2/2, .
d a lf>
\V f e shall prove that we may surround any 91-way space
with a space for which
(^')ft = 0; (r-i-) h = 0. (173.1)
** = -%
then A/, //>fc = (* / n H - {rw, n ti - { ikt } n lr )
\rkt} -n t , {Hot}.
A PARTICULAR SURROUNDING SPACE 251
Now we have seen that
and therefore
that is, A*.* = x * (^A-fc-^A-'Vfc)
= 0. (173,2)
Thus we see that the relations fl ri .^ = -^iVi- persist when
U. and lO row in accordance with the laws
+a x "/2,. x /2^=0, (173.3)
7^ + 2^=0. (173.4)
We can therefore surround the 7i-way space with a space
for which ( r ]>i.) b _ o ; (r-i-) b = 0.
174. Curvature properties of this surrounding space.
We will now consider what properties such a surrounding
space would have as regards curvature.
Consider the ground form of the surrounding space, which
we denote by the suffix ft,
Let B
7; = i;x? l + ... + v
be two vectors of lengths j | and | rj \ inclined at an angle &
which lie in the tangential (^+l)-fold and therefore in
Euclidean space.
Let = i 0i + ...+ 5
be two corresponding vectors of lengths | | and | TJ \ inclined
at an angle 6 aad lying in^he tangential n-fold.
252 71- WAY SPACE AS A LOCUS IN (+l)-\VAY SPACE
Wo have I I* = ''*,
4 | | 2 ! r, | sin* = (V-|V) (V-V) (-% V-
The measure of curvature 7f a , according to Riemann, which
corresponds to the orientation given by the vectors and ?;,
satisfies the equation
4 sin* 6 | | 2 h | 2 A' a - ( V- V) (V -
Now consider the vectors , 77 when u = 0.
i;| cos*)
= 0,
We therefore have
snce
But (
aud therefore
n pk - n ilt n M ). (174.1)
Here K b is the Riemann curvature in the (ra-f l)-way space
corresponding to the orientation of the vectors + z and
rj 4- 772; ; and iT a is the Riemann curvature in the '/i-way space
corresponding to the vectors and 77.
175. We may express the result in yet another form.
Consider the ground form cfl^dx^dx^y where c is a constant
introduced to keep the dimensions right, and let a vector (
be defined by the equations /.A.-f r/2-/ = 0.
The vector will then trace out in some Euclidean r-fold an
7i- way space vc l ...x n . * c
CURVATURE IN THE PARTICULAR SURROUNDING SPACE 253
In this space let us consider two vectors and 77 of lengths
| | and 77 | inclined at an angle 0', where
We have
4sin 2 0' | 2 1 77 1 2
= *('y-V)(W
and therefore
2 !r;i 2 + >) 2 !| 2 -277!!|77 cos0) = 0.
(175.1)
We see that the curvature of this (71+!)- way space which
surrounds the given ?i-way space depends, then, on the know-
ledge of the ground form fl^dx^dx^ with the property that
176. A condition that the surrounding space may be
Euclidean. We now ask whether the surrounding space can
be Euclidean ?
If it is Euclidean we must have
. 0.
We have seen in 173 that, if when u the equations
H ri . ll = fl rk . i ^ (176.1)
hold, they will persist for any value of u whilst n ik and a,-/,
grow in accordance with the laws
\^
(170.2)
= 0. (176.3)
254 7/,-WAY SPACE AS A LOCUS IN (u + l)-WAY SPACE
We shall now prove that if these equations hold, then, if
(rkih)+n rh n ik -n ri n hk - o (ire . 4)
holds when u = 0, it also will persist when u has any value.
The expression
~ (rkih) + {rtih} n ]d - {Idih} fl fl
is a tensor component : when we refer to the geodesic coordi-
nates of any given point we shall find that it vanishes at that
given point and therefore vanishes identically.
So referred,
(rkih) = - ( (r ik) (
ii i
Now = Sl fi . k + { rkt } n u + { ild} n, t ,
and therefore
~ (rkih) = A ( [ikt}il rt - { irt] n M )
= f2 rt [ktlh] -n l:l (rtih], (176 . 5)
which proves the required formula.
Again
fX a^ (kphi)
CONDITIONS FOR A EUCLIDEAN SURROUNDING SPACE 255
It follows that
^ ((rkih) + n^n^-n,^) = o ;
that is, the equations
Sl ik '-fl ti fl Ilk = 0, (176 . 6)
if true when -u = 0, will always be true.
The condition that an ')i-way space may be contained in
a Euclidean (H + l)-\vay space is that the equations
n ri . jf = n tjf . i9 (17G.7)
may be consistent.
177. Procedure for applying the condition when n > 2.
There is now an essential distinction between the case n 2
and the case n > 2.
A two-way space is always contained in a Euclidean space
of three dimensions, and we have considered the problems
associated with this case.
If 71 > 2 we can uniquely determine the functions fl ik in
terms of the four-index symbols of ChrisloiFel, by aid of the
equations
alone. If n > 3 we even have relations between the four-
index symbols from the consistency of these equations. It is
a problem of algebra merely to determine the functions /J^,
and the functions so determined arc tensor components.
If the surrounding space is to be Euclidean, the functions so
determined must satisfy the equations f2 t ^.j. = f2 r j { .^ We
can therefore, when we are given the ground form ^^t/.r^fo/.,
determine, by algebraic work merely, whether the space to
which the ground form refers is or is not contained within
a Euclidean (91+!)- way space. The actual work would,
however, be laborious.
178. The n-way space as a surface in the Euclidean
space when this exists. Suppose, ^iow, that we are given the
ground form a^dx^lxj^ and that we have found that the space
256 n-WAY SPACE AS A LOCUS IN (tt+l)-WAY SPACE
to which it refers is contained in a Euclidean (?i+l)-way
space and have calculated the functions fl ik : we may ask,
what is the surface in Euclidean space which is the given
n- way space?
Let z bo the vector in the Euclidean (n-fl)-way space
which traces out the given ?i-way space. We know from our
earlier work that z. ik is normal to each element of the space
drawn through the extremity of z. Now there is only one
such vector in the Euclidean (?i+ l)-fold. Let A be the unit
vector which is normal to the surface. Then
-to = ./, A, (178.1)
where w^ is some scalar.
We have ~ ., _ w .\. 4-?/> . , \
" ~
and therefore, since
kl-lk - \tpik\ (p- \qtik\ (J),
where p is an upper integer and q a lower integer, we have
- [Ttik\ Z t = W f i\k-Wrk*i + (Wri-k-- w rk*i)*- ( l 7S ' 2 >
Multiplying by A, and taking the scalar product, and noting
that X\ p = 0, we have v , f . fc = Wrk ^
We also have - = (rkhi),
and therefore w rh w ki~~~ w ri' w kh ~
It follows that w ik = /2 {/ ., (178 . 3)
and we have z. ik /2 l7 ,A. (178. 4)
We also have [rtik\ z t = /2 / . 7i .A i -/2 /{ A / ., (178 . 5)
so that when we know A we can find z by quadrature and
thus determine the surface save for a translation.
179. We have now to show how to determine A.
Xz p = 0,
and therefore ^g^ + Xz. 0.
It follows that XjZfr = t\^ = fl ik .
^~ ^- t
W-WAY SURFACE IN THE (tt + l)-WAY EUCLIDEAN SPACE 257
From the equation
{rtik} z t = fl fl6 \i-Sl ri
{rtik} fl tp = n rlc \i\ p - fl
and therefore
tt (rryi/c) /2, p = n, rk \j\ p -{2 H \ k \ l)
that LS, c*'tf/2, p (fi ri f2 qk -fi rk f2 qi ) = ^x^-^x^,
or /2,,. fe (AjA + a'S-fyp/fy) = ^(X + c^/2/2). (179 . 2)
Unless, then, the coefficients of f2 rk and /2^ are zero, we must
have n rk n gi = n ri n sk ,
which would mean that (rski) = and that the 76-way space
was Euclidean, a case we need not consider. We conclude that
\i\ k + aP ( in, pi n qk =0. (179 . 3)
We thus know the ground form of the surface traced out
by the unit vector X.
Let \. ik = ^~ -(ikt>'^> (179.4)
** *xt*x k <>^ t
where {ikt\* is formed witli reference to this ground form.
We have AX; = 0,
and therefore XX.^-hX^X^. 0.
Now \. ik is parallel to the normal to the surface traced out
by X, and therefoi*e, as X is a unit vector, is parallel to X.
it follows that A.^-ha^/i^^X = 0. (179 . 5)
We thus have the equations which determine X.
These equations may be written
A-<* + a '<*A = > (179.6)
where a' ik denotes a coefficient in the ground form of X. As
this ground form is that of a space of constant positive
curvature we see that the system is c complete '.
It follows that we can allow X t . . . X n to take any initial
values and thus we can determine X save as to a ' movement '
in Euclidean space.
258 ft- WAY SPACE AS A LOCUS IN (n + 1>WAY SPACE
We have considered three ground forms ; these maybe written
dzdz =
dzd\ =
d\d\ =
We saw (147.4) that the lines of principal curvature were
given by the equations (a pq -Rfl pq } dx q = 0,
that is now, by z p (dz Rd\) = 0,
and as we also have X (dz RdX) = 0,
^ ^/
we conclude that dz = Rd\ (179.7)
is the equation of the line of curvature corresponding to the
principal radius of curvature It.
INDEX
(The references are to pages)
Applicable surfaces, defined, 27,
determination of, 67.
examples of, 26, 28, 31.
Associated vector spaces, 1 7.
Asymptotic lines, reference to,
77, 79.
on minimal surfaces, 153.
Beltrami's differential parameters,
16, 17, 56, 80.
are invariants, 17.
Bertrand curves, 116.
Bianchi, 132, 171.
Bonnet's theorems on ruled sur-
faces, 131, 139.
Central surface of a congruence,
97.
Christoffel's three-index symbols,
6, 7, 56.
four-index symbols (tensor
components), 19, 20, 21.
Codazzi's fundamental equations,
64, 65, 67, 77, 132, 135.
moving axes equations, 113,
125.
Complex functions of position, 59.
Confbrmal representation, 155,
157.
on triangular area, 158, &c.
on a given polygon, 168.
Congruences of straight lines,
93, &c.
of normals, 104, 107.
Conjugate harmonic functions, 60.
lines with equal invariants,
83, 92.
Constant Riemauu curvature,
232.
Contracted notation, 1, 4, 6, 8,
12, 62.
Curvature and torsion of a curve,
110, 113, 180.
Curvature, lines of, 66, 68, 76,
105, 153, 179.
measure of, 39, 49, 64, 69, 181.
principal ladii of, 64, 230.
principal (rc-way), 208, 209,
219,
Curved space, the idea of, 2.
Darboux, 40, 56, 113, 171.
Derivatives of a ik , &c., 8.
Developable surfaces, 30, 68.
1 Erection cosines, extended notion
of, 5, 76.
Dummy suffixes, 6.
Einstein space, 23, 212.
extended, 24, 247.
four- way stationary, 228.
(n+ l)-way stationary, 220, &c.
(n+ l)-way surroundings- way,
207, 212.
Elliptic coordinates, 31.
Enneper's theorem, 125.
Equivalent ground forms, 32.
Euclidean coordinates at a point,
18, 21, 46, 190.
surrounding space, 2, 207, 253,
255,
Focal coordinates, 55, 72.
lines non-existent, 100.
glanes and points, 95.
surface of a congruence, 98,
105.
260
INDEX
Four-way stationary Einstein
space, 228.
Fundamental and associate ten-
sors, 14.
magnitudes, the two-way Q n ,
12 12 , M> 63, 65.
magnitudes, the w-way 12^,
207, 212.
Galilean coordinates at a point,
193.
Gauss, 65, 194, 198.
Geodesies, 42, 43, 54, 237.
in ra-way space, 188, 199, 203.
in w-way space of curvature
+ 1, 236.
Geodesic curvature, 45, 51, 53,
119, 208.
distance (w-way), 238.
equations integrable when K
is constant, 54.
orthogonal trajectories, 44.
polar coordinates, 192.
torsion, 120.
Ground form, with K constant,
34, 36.
with A(#) =0, 37.
with A (K) and A 2 (/r) functions
of K, 37, 38.
general two-way, 39.
Hamiltoniau equation, 96.
Harmonic functions, 60.
Helicoid, 117.
Henneberg's surface, 151.
Hypergcometric series, 158.
Integration of geodesic equations
with constant /C 54.
Invariant K, 32, 56, 64.
Invariants, systems of, 23, 24.
Isotropic congruences, 102.
the ruled surface of, 141.
Laplace, 83.
Lie, 108. ,
Limiting points of ray of con-
gruence, 96.
Lines in three-way space of curva-
ture + 1, 243.
Liouville, 58.
surfaces, 228.
Malus's theorem, 101.
Measure of curvature (see Curva-
ture, and Invariant K).
Riemann's oriented hi ^-way
space, 194, 198.
Mercator's projection, 61.
Mourner's theorem, 120.
Minimal surfaces, 70, 92, 103,
143, &c., 155.
associate and adjoint, 153.
double, 151.
Moving axes, 125.
n-way principal curvatures :
orthogonal directions of, 208,
229.
sum of products of, 209.
n- way space :
in a higher Euclidean space,
4, 17.
in (w-f l)-way space, 206, 250.
in (?i-f l)-way Einstein space,
212, 219.
111(71+ l)-way Euclidean space,
253, &c.
of constant curvature for all
orientations, 232, 247.
of zei o Kiemanu curvature, 231.
Normal vector, 17, 63, (>5, 78.
Null curves, 147.
Null lines, 58, 145.
Orthogonal burfaces, 127, 172, c.
intersect in lines of curvature,
179.
special classes of, 185, 187.
Parallel curves, 51.
Plateau, the problem of. 155, 168.
Polar coordinates in n-way space
of curvature + 1, 239.
geodesic coordinates, 47.
INDEX
261
Principal planes of a lay, 97.
surfaces of a congruence, 97.
Pseudosphere, 27, 75.
Pseudo-spherical trigonometry, 2 7.
Quaternion, 174.
notation, 62.
Rank of a tensor, 15.
Reduction of a two-way form :
with constant K, 34, &c.
with &(K) = 0, 36.
with A(]r) and A 2 (#) functions
of J{, 37.
in general, 38.
Refraction of a congruence, 1 00.
Regular solids, 167.
Ribaucour, 93, 101, 102.
Riccati's equation utilized, 39,
41, 108, 109, 115, &c.
Riemann, 171, 194.
Rotation functions, 40, 111, 178.
vector, 87.
Ruled surfaces, 28, 129, &c.
anharmonic properties on, 138,
139.
applicability on, 133, 135.
ground form for. 130.
line of stiiction on, 137.
which cut at one angle along
a generator, 140, 142.
Scalar product, notation for, 4, 62.
Schwartz, 171.
Second derivatives in transforma-
tion theory, 1 1.
Self-conjugate null curves, 148,
151. -T-
j/iQiyl i' P'l'i'n' 1 1 ii 1 1 and torsion
formulae, 110, 114.
Space of curvature -f 1, 237, 240.
of curvature zero, 231.
Spherical image, 66, 73.
trigonometry, 27.
Surfaces of revolution, applica-
bility on, 26, 107.
orthogonal to geodesies, 199,
203.
with plane lines of curvature,
122.
Tangential equations, 69.
Tangential Euclidean n-fold, 191.
Tensor calculus, rules, &c. of, 14.
components, defined, 9.
generated from others, 21.
derivatives, 12, 13, 16.
Tensors, fundamental and asso-
ciate, 14.
Three-way space of curvature + 1 ,
240, &c.
Tractrix, 28.
Translation surfaces, 150.
Twelve associated surfaces, 89, 9 1 .
Upper and lower integers, 7.
Vanishing invariants of Einstein
space, 23.
Vector, the rotated qOCq~ l , 174.
of triply orthogonal svstem,
178.
Vectors in a Euclidean space, 4.
ets of orthogonal, 175.
Vector-products, notation for, 62.
Volume in three-way space of
curvature -f 1, 246.
Voss, 208.
W congruences, 103, 106, 107.
W surfaces, 71, 72, 106.
-Woki^rass, 171.
Weingarten,""71
Zero Riemann curvature, space
of, 231. .
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